# Question 14755: Finite Convolution May Lose Degree over a Ring with Zero Divisors [Return to catalogue page 74](https://ocme.evemisslab.com/en/questions/page/74/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations/#q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0028-14755 · q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055 ## Original question and givens Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 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. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors). - G001 — Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 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. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-polynomial-ring-over-finite-commutative-ring](https://ocme.evemisslab.com/en/mko/mko-polynomial-ring-over-finite-commutative-ring/) ## Step-by-step derivation S001 · impliesBefore same carrier Fin(4);addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]];multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] After actual additive zero=0;actual multiplicative one=3;additive inverses=[0, 1, 3, 2] Conditions: none Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1. Step identifiers and dependenciesRule: commutative_parent_actual_tables_and_identities · Uses: G001 S002 · impliesBefore all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null] After additive commute=Yes;multiplicative commute=Yes;zero differs from one=Yes Conditions: none Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes. Step identifiers and dependenciesRule: complete_commutative_unital_parent_qualification · Uses: G001, S001 S003 · impliesBefore left polynomial:raw ascending coefficient labels=[3, 2, 0, 0];actual ring zero=0 After normalized finite support=[3, 2];last-index degree field=1 Conditions: none Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention. Step identifiers and dependenciesRule: polynomial_trailing_actual_zero_protocol · Uses: G001, S002 S004 · impliesBefore right polynomial:raw ascending coefficient labels=[1, 3, 0, 0];actual ring zero=0 After normalized finite support=[1, 3];last-index degree field=1 Conditions: none Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention. Step identifiers and dependenciesRule: polynomial_trailing_actual_zero_protocol · Uses: G001, S003 S005 · impliesBefore convolution coefficient0 times coefficient0:operation=multiplication;left label=3;right label=1;original row=[0, 1, 2, 3] After result label=1 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S004 S006 · impliesBefore accumulate product at degree0:operation=addition;left label=0;right label=1;original row=[0, 1, 2, 3] After result label=1 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S005 S007 · impliesBefore convolution coefficient0 times coefficient1:operation=multiplication;left label=3;right label=3;original row=[0, 1, 2, 3] After result label=3 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S006 S008 · impliesBefore accumulate product at degree1:operation=addition;left label=0;right label=3;original row=[0, 1, 2, 3] After result label=3 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S007 S009 · impliesBefore convolution coefficient1 times coefficient0:operation=multiplication;left label=2;right label=1;original row=[0, 1, 3, 2] After result label=1 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S008 S010 · impliesBefore accumulate product at degree1:operation=addition;left label=3;right label=1;original row=[3, 2, 0, 1] After result label=2 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S009 S011 · impliesBefore convolution coefficient1 times coefficient1:operation=multiplication;left label=2;right label=3;original row=[0, 1, 3, 2] After result label=2 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S010 S012 · impliesBefore accumulate product at degree2:operation=addition;left label=0;right label=2;original row=[0, 1, 2, 3] After result label=2 Conditions: none Use the original operation/order lookup.Labels are not integer sums/products,and modular guesses do not define the parent ring. Step identifiers and dependenciesRule: same_parent_operation_not_integer_labels · Uses: G001, S011 S013 · impliesBefore product result:raw ascending coefficient labels=[1, 2, 2];actual ring zero=0 After normalized finite support=[1, 2, 2];last-index degree field=2 Conditions: none Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention. Step identifiers and dependenciesRule: polynomial_trailing_actual_zero_protocol · Uses: G001, S012 S014 · impliesBefore normalized product=[1, 2, 2];last-index degree field=2 After degree_additivity_not_claimed_with_zero_divisors Conditions: none Convolve finite supports using parent-ring operations,then trim trailing zeros.Leading coefficients may annihilate through zero divisors,and two nonzero polynomials may multiply to zero.No unconditional degree-additivity or field property is claimed. Step identifiers and dependenciesRule: polynomial_convolution_zero_divisor_boundary · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c3ab9da82bec328a8afb1b5a49c314ca6c27cc973a19594ecf4165a423eb36a8 - candidate record SHA-256: 4d85fc8027ba7a51ecc39c07ec9f12ebbb916e414db2538527d9c7c321f743c0 - semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae - question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055.json)