Complete Zero-Divisor Audits, Nonzero Cancellation and the Finite Inverse Step
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batch-20261014-advance-20261029-r01-finite-integral-domain-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(4, 0, 1, [[2, 2]], No, commutative_ring_with_zero_divisors)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (4, 0, 1, [[2, 2]], No, commutative_ring_with_zero_divisors).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
2. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 1, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 1, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
3. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 1, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 1, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
4. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 1, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 1, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
5. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 1, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 1, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
6. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(4, 0, 3, [[2, 2]], No, commutative_ring_with_zero_divisors)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (4, 0, 3, [[2, 2]], No, commutative_ring_with_zero_divisors).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
7. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 2, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 2, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
8. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 2, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 2, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
9. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 2, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 2, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
10. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 2, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 2, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
11. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(4, 1, 3, [[0, 0]], No, commutative_ring_with_zero_divisors)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (4, 1, 3, [[0, 0]], No, commutative_ring_with_zero_divisors).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
12. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 2, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 2, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
13. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 3, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 3, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
14. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 3, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 3, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
15. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 3, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 3, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
16. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(4, 2, 1, [[3, 3]], No, commutative_ring_with_zero_divisors)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (4, 2, 1, [[3, 3]], No, commutative_ring_with_zero_divisors).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
17. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 3, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 3, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
18. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 3, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 3, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
19. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 4, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 4, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
20. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 4, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 4, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
21. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(4, 3, 1, [[0, 0]], No, commutative_ring_with_zero_divisors)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (4, 3, 1, [[0, 0]], No, commutative_ring_with_zero_divisors).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
22. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 4, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 4, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
23. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 4, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 4, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
24. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 0, 4, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 0, 4, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
25. An Integral Domain Requires Every Nonzero Pair to Avoid Zero
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (n,zero,one,zero_divisor_pairs,domain?,classification). List all ordered nonzero [a,b] with a*b=zero in increasing a then b order. Classification is finite_integral_domain or commutative_ring_with_zero_divisors.
View answer and explanation
(5, 1, 0, [], Yes, finite_integral_domain)
Use the actual additive zero, not its presumed label. A complete absence of such pairs qualifies this finite integral domain. Invertibility is a subsequent theorem using finiteness, not an extra definition clause. The exact result, retaining the requested order, is (5, 1, 0, [], Yes, finite_integral_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-qualification · candidate
26. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
27. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
28. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
29. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
30. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
31. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
32. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
33. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
34. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
35. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
36. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
37. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
38. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
39. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
40. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
41. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
42. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
43. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
44. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
45. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
46. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
47. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
48. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
49. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
50. Audit All Nonzero Zero-Divisor Pairs
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Give (zero_divisor_pairs,no_nonzero_zero_divisors?), with pairs in increasing first- then second-label order.
View answer and explanation
([], Yes)
Search every ordered pair of nonzero elements in the complete multiplication table. A single pair yielding zero refutes the no-zero-divisor obligation; sampled successes do not discharge it. The exact result, retaining the requested order, is ([], Yes).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-zero-divisor-audit · candidate
51. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=1, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 1, 2, 1, 2, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 1, 2, 1, 2, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
52. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 2, 2, 3, 3, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 2, 2, 3, 3, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
53. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 2, 2, 4, 4, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 2, 2, 4, 4, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
54. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
55. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=0, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 0, 1, 0, 1, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 0, 1, 0, 1, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
56. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=4, with b=1, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(4, 1, 1, 4, 4, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (4, 1, 1, 4, 4, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
57. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=3, with b=1, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(3, 1, 1, 2, 2, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 1, 1, 2, 2, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
58. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=3, with b=4, c=4. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(3, 4, 4, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 4, 4, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
59. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=4, c=0. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 4, 0, 4, 0, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 4, 0, 4, 0, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
60. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=3, with b=0, c=0. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(3, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
61. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
62. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=4, c=4. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 4, 4, 3, 3, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 4, 4, 3, 3, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
63. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=3, with b=2, c=4. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(3, 2, 4, 2, 4, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 2, 4, 2, 4, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
64. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 2, 2, 4, 4, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 2, 2, 4, 4, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
65. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=0, c=0. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
66. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=3, c=3. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 3, 3, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 3, 3, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
67. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=3, with b=4, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(3, 4, 1, 4, 1, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (3, 4, 1, 4, 1, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
68. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=4, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(4, 2, 2, 3, 3, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (4, 2, 2, 3, 3, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
69. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=3, c=3. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 3, 3, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 3, 3, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
70. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=0, c=0. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 0, 0, 0, 0, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
71. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=4, with b=4, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(4, 4, 2, 4, 2, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (4, 4, 2, 4, 2, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
72. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=2, with b=2, c=2. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(2, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (2, 2, 2, 1, 1, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
73. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=1, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 1, 1, 2, 2, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 1, 1, 2, 2, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
74. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=1, with b=1, c=1. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(1, 1, 1, 4, 4, Yes, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (1, 1, 1, 4, 4, Yes, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
75. Nonzero Cancellation Has an Implication, Not an Equality Premise Flag
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require a qualified integral domain and nonzero multiplier a=0, with b=1, c=0. Give (a,b,c,ab,ac,premise_equal?,cancellation_valid?,nonzero_cancellation_in_domain).
View answer and explanation
(0, 1, 0, 1, 0, No, Yes, nonzero_cancellation_in_domain)
Compute ab and ac in the same table. cancellation_valid is the implication ab=ac implies b=c; it can be Yes even when the separately reported premise_equal is No. Zero multipliers and rings with nonzero zero divisors are rejected, not repaired. The exact result, retaining the requested order, is (0, 1, 0, 1, 0, No, Yes, nonzero_cancellation_in_domain).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-cancellation · candidate
76. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 1, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 1, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
77. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 1, 4, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 1, 4, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
78. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 4, 1, 3], [0, 3, 1, 4, 2], [0, 4, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 1, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 1, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
79. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 1, 2, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 1, 2, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
80. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 1, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 1, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
81. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=4. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(4, 1, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (4, 1, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
82. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=3. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(3, 2, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (3, 2, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
83. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=3. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(3, 2, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (3, 2, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
84. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 2, 2, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 2, 2, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
85. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=3. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(3, 2, 4, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (3, 2, 4, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
86. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 2, 4, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 2, 4, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
87. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 2, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 2, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
88. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=3. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(3, 3, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (3, 3, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
89. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 3, 4, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 3, 4, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
90. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 1, 2], [0, 3, 4, 2, 1], [0, 1, 2, 3, 4], [0, 2, 1, 4, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 3, 2, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 3, 2, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
91. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 3, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 3, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
92. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 4, 1, 3], [0, 4, 3, 2, 1], [0, 1, 2, 3, 4], [0, 3, 1, 4, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=3. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(3, 3, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (3, 3, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
93. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=4. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(4, 3, 2, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (4, 3, 2, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
94. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 4, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 4, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
95. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 4, 2, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 4, 2, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
96. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=4. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(4, 4, 4, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (4, 4, 4, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
97. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=2. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(2, 4, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (2, 4, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
98. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 4, 3, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 4, 3, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
99. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 3, 2, 1], [0, 3, 1, 4, 2], [0, 2, 4, 1, 3], [0, 1, 2, 3, 4]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=1. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(1, 4, 1, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (1, 4, 1, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
100. Keep the Finite Step from Integral Domain to Invertibility
Use the same qualified complete finite commutative unital ring on Fin(5)={0,…,4}, with addition=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] and multiplication=[[0, 1, 2, 3, 4], [1, 1, 1, 1, 1], [2, 1, 4, 0, 3], [3, 1, 0, 4, 2], [4, 1, 3, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Additionally require an integral domain and nonzero element=0. Give (element,one,inverse,found?,finite_injective_multiplication_implies_surjective).
View answer and explanation
(0, 0, 0, Yes, finite_injective_multiplication_implies_surjective)
Multiplication by a nonzero element is injective by cancellation. Because this carrier is finite it is also surjective, so the actual identity has a preimage. Return the first inverse in increasing carrier-label order; do not extend this finite step to arbitrary infinite domains. The exact result, retaining the requested order, is (0, 0, 0, Yes, finite_injective_multiplication_implies_surjective).
Reviewed step-by-step derivation
Finite integral domain · qf-mod29-integraldomain-finite-inverse · candidate
Canonical question JSONL · English JSONL · Validation report