Question 13547: A Power-of-Product Identity in a Qualified Commutative Ring

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0026-13547 · q-20261014-advance-20261028-r01-finite-commutative-ring-0072

Original question and givens

Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require multiplication to commute. a=6, b=1, nonnegative exponent=1. Give (a,b,exponent,ab,lhs=(ab)^exponent,rhs=a^exponent*b^exponent,equal?).

G001 — Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require multiplication to commute. a=6, b=1, nonnegative exponent=1. Give (a,b,exponent,ab,lhs=(ab)^exponent,rhs=a^exponent*b^exponent,equal?).

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-finite-commutative-ring

Step-by-step derivation

S001 · implies

Before

same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 7, 0, 6, 4, 5], [2, 3, 7, 5, 1, 4, 0, 6], [3, 7, 5, 6, 2, 0, 1, 4], [4, 0, 1, 2, 6, 7, 5, 3], [5, 6, 4, 0, 7, 2, 3, 1], [6, 4, 0, 1, 5, 3, 7, 2], [7, 5, 6, 4, 3, 1, 2, 0]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 6, 6, 2, 7, 0], [0, 3, 6, 1, 5, 4, 2, 7], [0, 4, 6, 5, 1, 3, 2, 7], [0, 5, 2, 4, 3, 1, 6, 7], [0, 6, 7, 2, 2, 6, 7, 0], [0, 7, 0, 7, 7, 7, 0, 0]]

After

both complete operation tables typed and closed

Conditions: none

Bind subsequent ideals or commutativity to this same parent ring.Preserve the complete original tables;carrier labels are not integer arithmetic values.

Step identifiers and dependencies

Rule: parent_ring_original_complete_tables · Uses: G001

S002 · implies

Before

every additive triple checked;first associativity failure=null;all additive pairs commute=Yes

After

additive zero candidates=[0];actual zero=0;two-sided additive inverses in label order=[0, 4, 6, 5, 1, 3, 2, 7]

Conditions: none

Audit additive associativity,commutativity,the two-sided zero and each inverse over the complete carrier.This qualified parent is the problem's premise,checked by complete finite conditions rather than samples.

Step identifiers and dependencies

Rule: parent_additive_abelian_group_complete_audit · Uses: G001, S001

S003 · implies

Before

every multiplicative triple checked;first associativity failure=null

After

multiplicative identity candidates=[1];actual one=1

Conditions: none

Multiplication here requires associativity and a two-sided one,not commutativity or inverses of every nonzero element.

Step identifiers and dependencies

Rule: parent_multiplicative_monoid_complete_audit · Uses: G001, S002

S004 · implies

Before

all 512 left and 512 right distributivity triples checked

After

first left failure=null;first right failure=null;zero=0;one=1;distinct=Yes

Conditions: none

Audit both distributive obligations completely and separately.Zero and one must be distinct actual elements.Unital alone implies neither a commutative ring,domain nor field.

Step identifiers and dependencies

Rule: parent_both_distributivities_and_distinct_identities · Uses: G001, S003

S005 · implies

Before

unordered parent elements=[0, 1];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S004

S006 · implies

Before

unordered parent elements=[0, 2];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S005

S007 · implies

Before

unordered parent elements=[0, 3];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S006

S008 · implies

Before

unordered parent elements=[0, 4];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S007

S009 · implies

Before

unordered parent elements=[0, 5];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S008

S010 · implies

Before

unordered parent elements=[0, 6];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S009

S011 · implies

Before

unordered parent elements=[0, 7];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S010

S012 · implies

Before

unordered parent elements=[1, 2];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S011

S013 · implies

Before

unordered parent elements=[1, 3];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S012

S014 · implies

Before

unordered parent elements=[1, 4];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S013

S015 · implies

Before

unordered parent elements=[1, 5];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S014

S016 · implies

Before

unordered parent elements=[1, 6];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S015

S017 · implies

Before

unordered parent elements=[1, 7];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S016

S018 · implies

Before

unordered parent elements=[2, 3];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S017

S019 · implies

Before

unordered parent elements=[2, 4];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S018

S020 · implies

Before

unordered parent elements=[2, 5];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S019

S021 · implies

Before

unordered parent elements=[2, 6];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S020

S022 · implies

Before

unordered parent elements=[2, 7];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S021

S023 · implies

Before

unordered parent elements=[3, 4];a*b=5;b*a=5

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S022

S024 · implies

Before

unordered parent elements=[3, 5];a*b=4;b*a=4

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S023

S025 · implies

Before

unordered parent elements=[3, 6];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S024

S026 · implies

Before

unordered parent elements=[3, 7];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S025

S027 · implies

Before

unordered parent elements=[4, 5];a*b=3;b*a=3

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S026

S028 · implies

Before

unordered parent elements=[4, 6];a*b=2;b*a=2

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S027

S029 · implies

Before

unordered parent elements=[4, 7];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S028

S030 · implies

Before

unordered parent elements=[5, 6];a*b=6;b*a=6

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S029

S031 · implies

Before

unordered parent elements=[5, 7];a*b=7;b*a=7

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S030

S032 · implies

Before

unordered parent elements=[6, 7];a*b=0;b*a=0

After

commute=Yes

Conditions: none

Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity.

Step identifiers and dependencies

Rule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S031

S033 · implies

Before

all multiplicative noncommuting records=[]

After

parent ring commutative=Yes;zero divisors not excluded

Conditions: none

This adds multiplicative commutativity only.It excludes neither zero divisors nor lack of multiplicative inverses.

Step identifiers and dependencies

Rule: commutative_ring_extra_obligation_not_field · Uses: G001, S032

S034 · implies

Before

a times b:operation=multiplication;left label=6;right label=1;table row=[0, 6, 7, 2, 2, 6, 7, 0]

After

result label=6

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S033

S035 · implies

Before

product power:base label=6;nonnegative exponent=1;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S034

S036 · implies

Before

product power step1:operation=multiplication;left label=1;right label=6;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=6

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S035

S037 · implies

Before

a power:base label=6;nonnegative exponent=1;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S036

S038 · implies

Before

a power step1:operation=multiplication;left label=1;right label=6;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=6

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S037

S039 · implies

Before

b power:base label=1;nonnegative exponent=1;actual one=1

After

initial power0=1

Conditions: none

Start every nonnegative power at the actual one,including exponent0;its label need not be1.

Step identifiers and dependencies

Rule: unital_ring_power_zero_boundary · Uses: G001, S038

S040 · implies

Before

b power step1:operation=multiplication;left label=1;right label=1;table row=[0, 1, 2, 3, 4, 5, 6, 7]

After

result label=1

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S039

S041 · implies

Before

product of powers:operation=multiplication;left label=6;right label=1;table row=[0, 6, 7, 2, 2, 6, 7, 0]

After

result label=6

Conditions: none

Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels.

Step identifiers and dependencies

Rule: parent_table_operation_lookup · Uses: G001, S040

S042 · implies

Before

whole parent multiplication commutes;exponent=1;lhs=6;rhs=6

After

(a*b)^k=a^k*b^k under associativity and commutativity,including k0

Conditions: none

Associativity and whole-ring commutativity permit regrouping the finite factors.At k=0 both sides are one.This instance is not used to establish commutativity.

Step identifiers and dependencies

Rule: commutative_power_product_rearrangement · Uses: G001, S041

Reverse checks

S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(6, 1, 1, 6, 6, 6, Yes)

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: 646dc18b0ca682756afef4cef8078b926fa253d2751cf1e51f4169c9a5f8e469
candidate record SHA-256: e4923fc3cda1939fe28cde8ebff10685d0bde7c8769db0880416c5ef2d5fda94
semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798
question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77
Lean: NOT_RUN · human review: NOT_PERFORMED

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