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Question 13746: Two-Sided Ideals Require Three Complete Qualifications

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0026-13746 · q-20261014-advance-20261028-r02-finite-two-sided-ideal-0021

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, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 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. Normalize subset=[0, 1, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.

  • 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, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 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. Normalize subset=[0, 1, 2, 4, 7] as distinct carrier labels in increasing order. Its operations and inverse map come from this same parent ring. Additive-subgroup qualification, left absorption and right absorption are independent obligations. No quotient, maximality or primality is constructed. Give (subset,zero,additive_subgroup?,missing_left,missing_right,two_sided_ideal?). Missing-left rows are [r,x,r*x]; missing-right rows are [x,r,x*r], ordered by parent label r then subset label x.

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.

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, 6, 4, 0, 5], [2, 3, 7, 5, 0, 6, 1, 4], [3, 7, 5, 4, 1, 0, 2, 6], [4, 6, 0, 1, 7, 3, 5, 2], [5, 4, 6, 0, 3, 2, 7, 1], [6, 0, 1, 2, 5, 7, 4, 3], [7, 5, 4, 6, 2, 1, 3, 0]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 7, 4, 7, 2, 4, 0], [0, 3, 4, 1, 2, 6, 5, 7], [0, 4, 7, 2, 7, 4, 2, 0], [0, 5, 2, 6, 4, 1, 3, 7], [0, 6, 4, 5, 2, 3, 1, 7], [0, 7, 0, 7, 0, 7, 7, 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, 6, 4, 5, 2, 3, 1, 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

raw subset=[0, 1, 2, 4, 7];same qualified parent zero=0

After

sorted distinct parent labels=[0, 1, 2, 4, 7]

Conditions: none

Normalize the subset as a sorted set,but obtain all operations and inverses from the same parent ring.Do not substitute new operations or integer arithmetic on labels.

Step identifiers and dependencies

Rule: ideal_subset_same_parent_normalization · Uses: G001, S004

S006 · implies

Before

subset element=0;parent additive inverse=0

After

inverse belongs to subset=Yes

Conditions: none

An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse.

Step identifiers and dependencies

Rule: ideal_each_parent_additive_inverse · Uses: G001, S005

S007 · implies

Before

subset ordered pair=[0, 0];parent sum=0

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S006

S008 · implies

Before

subset ordered pair=[0, 1];parent sum=1

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S007

S009 · implies

Before

subset ordered pair=[0, 2];parent sum=2

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S008

S010 · implies

Before

subset ordered pair=[0, 4];parent sum=4

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S009

S011 · implies

Before

subset ordered pair=[0, 7];parent sum=7

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S010

S012 · implies

Before

subset element=1;parent additive inverse=6

After

inverse belongs to subset=No

Conditions: none

An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse.

Step identifiers and dependencies

Rule: ideal_each_parent_additive_inverse · Uses: G001, S011

S013 · implies

Before

subset ordered pair=[1, 0];parent sum=1

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S012

S014 · implies

Before

subset ordered pair=[1, 1];parent sum=2

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S013

S015 · implies

Before

subset ordered pair=[1, 2];parent sum=3

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S014

S016 · implies

Before

subset ordered pair=[1, 4];parent sum=6

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S015

S017 · implies

Before

subset ordered pair=[1, 7];parent sum=5

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S016

S018 · implies

Before

subset element=2;parent additive inverse=4

After

inverse belongs to subset=Yes

Conditions: none

An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse.

Step identifiers and dependencies

Rule: ideal_each_parent_additive_inverse · Uses: G001, S017

S019 · implies

Before

subset ordered pair=[2, 0];parent sum=2

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S018

S020 · implies

Before

subset ordered pair=[2, 1];parent sum=3

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S019

S021 · implies

Before

subset ordered pair=[2, 2];parent sum=7

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S020

S022 · implies

Before

subset ordered pair=[2, 4];parent sum=0

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S021

S023 · implies

Before

subset ordered pair=[2, 7];parent sum=4

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S022

S024 · implies

Before

subset element=4;parent additive inverse=2

After

inverse belongs to subset=Yes

Conditions: none

An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse.

Step identifiers and dependencies

Rule: ideal_each_parent_additive_inverse · Uses: G001, S023

S025 · implies

Before

subset ordered pair=[4, 0];parent sum=4

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S024

S026 · implies

Before

subset ordered pair=[4, 1];parent sum=6

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S025

S027 · implies

Before

subset ordered pair=[4, 2];parent sum=0

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S026

S028 · implies

Before

subset ordered pair=[4, 4];parent sum=7

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S027

S029 · implies

Before

subset ordered pair=[4, 7];parent sum=2

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S028

S030 · implies

Before

subset element=7;parent additive inverse=7

After

inverse belongs to subset=Yes

Conditions: none

An additive subgroup needs each member's parent-ring additive inverse to remain in the subset,not a multiplicative inverse.

Step identifiers and dependencies

Rule: ideal_each_parent_additive_inverse · Uses: G001, S029

S031 · implies

Before

subset ordered pair=[7, 0];parent sum=7

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S030

S032 · implies

Before

subset ordered pair=[7, 1];parent sum=5

After

sum belongs to subset=No

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S031

S033 · implies

Before

subset ordered pair=[7, 2];parent sum=4

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S032

S034 · implies

Before

subset ordered pair=[7, 4];parent sum=2

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S033

S035 · implies

Before

subset ordered pair=[7, 7];parent sum=0

After

sum belongs to subset=Yes

Conditions: none

Check every ordered addition pair;sampling does not replace closure.

Step identifiers and dependencies

Rule: ideal_all_additive_closure_pairs · Uses: G001, S034

S036 · implies

Before

subset nonempty=Yes;actual zero belongs=Yes;missing sums=[[1, 2, 3], [1, 4, 6], [1, 7, 5], [2, 1, 3], [4, 1, 6], [7, 1, 5]];missing inverses=[[1, 6]]

After

additive subgroup=No

Conditions: none

Nonemptiness,the actual zero,addition closure and inverse closure must all hold.An empty subset does not become an additive subgroup through vacuous closure checks.

Step identifiers and dependencies

Rule: ideal_additive_subgroup_nonempty_zero_and_closure · Uses: G001, S035

S037 · implies

Before

parent r=0;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S036

S038 · implies

Before

parent r=0;subset x=1;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S037

S039 · implies

Before

parent r=0;subset x=2;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S038

S040 · implies

Before

parent r=0;subset x=4;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S039

S041 · implies

Before

parent r=0;subset x=7;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S040

S042 · implies

Before

parent r=1;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S041

S043 · implies

Before

parent r=1;subset x=1;left product r*x=1;right product x*r=1

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S042

S044 · implies

Before

parent r=1;subset x=2;left product r*x=2;right product x*r=2

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S043

S045 · implies

Before

parent r=1;subset x=4;left product r*x=4;right product x*r=4

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S044

S046 · implies

Before

parent r=1;subset x=7;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S045

S047 · implies

Before

parent r=2;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S046

S048 · implies

Before

parent r=2;subset x=1;left product r*x=2;right product x*r=2

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S047

S049 · implies

Before

parent r=2;subset x=2;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S048

S050 · implies

Before

parent r=2;subset x=4;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S049

S051 · implies

Before

parent r=2;subset x=7;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S050

S052 · implies

Before

parent r=3;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S051

S053 · implies

Before

parent r=3;subset x=1;left product r*x=3;right product x*r=3

After

left absorbed=No;right absorbed=No

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S052

S054 · implies

Before

parent r=3;subset x=2;left product r*x=4;right product x*r=4

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S053

S055 · implies

Before

parent r=3;subset x=4;left product r*x=2;right product x*r=2

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S054

S056 · implies

Before

parent r=3;subset x=7;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S055

S057 · implies

Before

parent r=4;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S056

S058 · implies

Before

parent r=4;subset x=1;left product r*x=4;right product x*r=4

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S057

S059 · implies

Before

parent r=4;subset x=2;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S058

S060 · implies

Before

parent r=4;subset x=4;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S059

S061 · implies

Before

parent r=4;subset x=7;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S060

S062 · implies

Before

parent r=5;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S061

S063 · implies

Before

parent r=5;subset x=1;left product r*x=5;right product x*r=5

After

left absorbed=No;right absorbed=No

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S062

S064 · implies

Before

parent r=5;subset x=2;left product r*x=2;right product x*r=2

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S063

S065 · implies

Before

parent r=5;subset x=4;left product r*x=4;right product x*r=4

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S064

S066 · implies

Before

parent r=5;subset x=7;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S065

S067 · implies

Before

parent r=6;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S066

S068 · implies

Before

parent r=6;subset x=1;left product r*x=6;right product x*r=6

After

left absorbed=No;right absorbed=No

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S067

S069 · implies

Before

parent r=6;subset x=2;left product r*x=4;right product x*r=4

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S068

S070 · implies

Before

parent r=6;subset x=4;left product r*x=2;right product x*r=2

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S069

S071 · implies

Before

parent r=6;subset x=7;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S070

S072 · implies

Before

parent r=7;subset x=0;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S071

S073 · implies

Before

parent r=7;subset x=1;left product r*x=7;right product x*r=7

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S072

S074 · implies

Before

parent r=7;subset x=2;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S073

S075 · implies

Before

parent r=7;subset x=4;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S074

S076 · implies

Before

parent r=7;subset x=7;left product r*x=0;right product x*r=0

After

left absorbed=Yes;right absorbed=Yes

Conditions: none

Test both sides for every parent r and subset x.A noncommutative parent may pass only one side.Preserve parent-r outer/subset-x inner order in failure lists.

Step identifiers and dependencies

Rule: ideal_both_absorptions_all_parent_elements · Uses: G001, S075

S077 · implies

Before

missing left [r,x,rx]=[[3, 1, 3], [5, 1, 5], [6, 1, 6]];missing right [x,r,xr]=[[1, 3, 3], [1, 5, 5], [1, 6, 6]]

After

left absorbing=No;right absorbing=No;empty subset absorption is vacuous

Conditions: none

Retain both absorption obligations separately.An empty subset can pass them vacuously without becoming an additive subgroup or ideal.

Step identifiers and dependencies

Rule: ideal_absorption_not_additive_subgroup · Uses: G001, S076

S078 · implies

Before

additive subgroup=No;left absorbing=No;right absorbing=No

After

two-sided ideal=No

Conditions: none

Conjoin additive subgroup,left absorption and right absorption.A subring or one-sided ideal is not automatically a two-sided ideal.

Step identifiers and dependencies

Rule: ideal_three_complete_obligations_conjunction · Uses: G001, S077

Reverse checks

  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

([0, 1, 2, 4, 7], 0, No, [[3, 1, 3], [5, 1, 5], [6, 1, 6]], [[1, 3, 3], [1, 5, 5], [1, 6, 6]], No)

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: 3018dd5e0f218d7a98b06ffa0a88d22a630ea7a2c410650e69712d6a275ea2ae
  • candidate record SHA-256: d21c4ba9f84122a2955ab1f21331027c15b7b4855fc7812f60dad05362ec48c8
  • semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798
  • question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON