# Question 13768: The Zero Ideal and Whole Ring Are Not the Empty Subset [Return to catalogue page 69](https://ocme.evemisslab.com/en/questions/page/69/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r02-finite-two-sided-ideal-foundations/#q-20261014-advance-20261028-r02-finite-two-sided-ideal-0093) 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. R0026-13768 · q-20261014-advance-20261028-r02-finite-two-sided-ideal-0093 ## 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, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] 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, 3, 4, 5, 6, 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_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals). - 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, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] 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, 3, 4, 5, 6, 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_ideal?,whole_ring?,qualified?,zero_and_whole_are_always_two_sided_ideals). ## 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-two-sided-ideal](https://ocme.evemisslab.com/en/mko/mko-finite-two-sided-ideal/) ## Step-by-step derivation S001 · impliesBefore same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 0, 4, 7, 5], [2, 3, 6, 7, 1, 0, 5, 4], [3, 6, 7, 5, 2, 1, 4, 0], [4, 0, 1, 2, 5, 7, 3, 6], [5, 4, 0, 1, 7, 6, 2, 3], [6, 7, 5, 4, 3, 2, 0, 1], [7, 5, 4, 0, 6, 3, 1, 2]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 5, 5, 6, 0, 2], [0, 3, 5, 1, 7, 2, 6, 4], [0, 4, 5, 7, 1, 2, 6, 3], [0, 5, 6, 2, 2, 6, 0, 5], [0, 6, 0, 6, 6, 0, 0, 6], [0, 7, 2, 4, 3, 5, 6, 1]] 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 dependenciesRule: parent_ring_original_complete_tables · Uses: G001 S002 · impliesBefore 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, 5, 7, 1, 2, 6, 3] 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 dependenciesRule: parent_additive_abelian_group_complete_audit · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: parent_multiplicative_monoid_complete_audit · Uses: G001, S002 S004 · impliesBefore 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 dependenciesRule: parent_both_distributivities_and_distinct_identities · Uses: G001, S003 S005 · impliesBefore raw subset=[0, 1, 2, 3, 4, 5, 6, 7];same qualified parent zero=0 After sorted distinct parent labels=[0, 1, 2, 3, 4, 5, 6, 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 dependenciesRule: ideal_subset_same_parent_normalization · Uses: G001, S004 S006 · impliesBefore 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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S005 S007 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S006 S008 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S007 S009 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S008 S010 · impliesBefore subset ordered pair=[0, 3];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S009 S011 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S010 S012 · impliesBefore subset ordered pair=[0, 5];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S011 S013 · impliesBefore subset ordered pair=[0, 6];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S012 S014 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S013 S015 · impliesBefore subset element=1;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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S014 S016 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S015 S017 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S016 S018 · impliesBefore subset ordered pair=[1, 2];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S017 S019 · impliesBefore subset ordered pair=[1, 3];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S018 S020 · impliesBefore subset ordered pair=[1, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S019 S021 · impliesBefore subset ordered pair=[1, 5];parent sum=4 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S020 S022 · impliesBefore subset ordered pair=[1, 6];parent sum=7 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S021 S023 · impliesBefore subset ordered pair=[1, 7];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S022 S024 · impliesBefore subset element=2;parent additive inverse=5 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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S023 S025 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S024 S026 · impliesBefore subset ordered pair=[2, 1];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S025 S027 · impliesBefore subset ordered pair=[2, 2];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S026 S028 · impliesBefore subset ordered pair=[2, 3];parent sum=7 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S027 S029 · impliesBefore subset ordered pair=[2, 4];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S028 S030 · impliesBefore subset ordered pair=[2, 5];parent sum=0 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S029 S031 · impliesBefore subset ordered pair=[2, 6];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S030 S032 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S031 S033 · impliesBefore subset element=3;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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S032 S034 · impliesBefore subset ordered pair=[3, 0];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S033 S035 · impliesBefore subset ordered pair=[3, 1];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S034 S036 · impliesBefore subset ordered pair=[3, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S035 S037 · impliesBefore subset ordered pair=[3, 3];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S036 S038 · impliesBefore subset ordered pair=[3, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S037 S039 · impliesBefore subset ordered pair=[3, 5];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S038 S040 · impliesBefore subset ordered pair=[3, 6];parent sum=4 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S039 S041 · impliesBefore subset ordered pair=[3, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S040 S042 · impliesBefore subset element=4;parent additive inverse=1 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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S041 S043 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S042 S044 · impliesBefore subset ordered pair=[4, 1];parent sum=0 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S043 S045 · impliesBefore subset ordered pair=[4, 2];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S044 S046 · impliesBefore subset ordered pair=[4, 3];parent sum=2 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S045 S047 · impliesBefore subset ordered pair=[4, 4];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S046 S048 · impliesBefore subset ordered pair=[4, 5];parent sum=7 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S047 S049 · impliesBefore subset ordered pair=[4, 6];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S048 S050 · impliesBefore subset ordered pair=[4, 7];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S049 S051 · impliesBefore subset element=5;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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S050 S052 · impliesBefore subset ordered pair=[5, 0];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S051 S053 · impliesBefore subset ordered pair=[5, 1];parent sum=4 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S052 S054 · impliesBefore subset ordered pair=[5, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S053 S055 · impliesBefore subset ordered pair=[5, 3];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S054 S056 · impliesBefore subset ordered pair=[5, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S055 S057 · impliesBefore subset ordered pair=[5, 5];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S056 S058 · impliesBefore subset ordered pair=[5, 6];parent sum=2 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S057 S059 · impliesBefore subset ordered pair=[5, 7];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S058 S060 · impliesBefore subset element=6;parent additive inverse=6 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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S059 S061 · impliesBefore subset ordered pair=[6, 0];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S060 S062 · impliesBefore subset ordered pair=[6, 1];parent sum=7 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S061 S063 · impliesBefore subset ordered pair=[6, 2];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S062 S064 · impliesBefore subset ordered pair=[6, 3];parent sum=4 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S063 S065 · impliesBefore subset ordered pair=[6, 4];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S064 S066 · impliesBefore subset ordered pair=[6, 5];parent sum=2 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S065 S067 · impliesBefore subset ordered pair=[6, 6];parent sum=0 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S066 S068 · impliesBefore subset ordered pair=[6, 7];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S067 S069 · impliesBefore subset element=7;parent additive inverse=3 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 dependenciesRule: ideal_each_parent_additive_inverse · Uses: G001, S068 S070 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S069 S071 · impliesBefore subset ordered pair=[7, 1];parent sum=5 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S070 S072 · impliesBefore 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S071 S073 · impliesBefore subset ordered pair=[7, 3];parent sum=0 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S072 S074 · impliesBefore subset ordered pair=[7, 4];parent sum=6 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S073 S075 · impliesBefore subset ordered pair=[7, 5];parent sum=3 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S074 S076 · impliesBefore subset ordered pair=[7, 6];parent sum=1 After sum belongs to subset=Yes Conditions: none Check every ordered addition pair;sampling does not replace closure. Step identifiers and dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S075 S077 · impliesBefore subset ordered pair=[7, 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 dependenciesRule: ideal_all_additive_closure_pairs · Uses: G001, S076 S078 · impliesBefore subset nonempty=Yes;actual zero belongs=Yes;missing sums=[];missing inverses=[] After additive subgroup=Yes 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 dependenciesRule: ideal_additive_subgroup_nonempty_zero_and_closure · Uses: G001, S077 S079 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S078 S080 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S079 S081 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S080 S082 · impliesBefore parent r=0;subset x=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S081 S083 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S082 S084 · impliesBefore parent r=0;subset x=5;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S083 S085 · impliesBefore parent r=0;subset x=6;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S084 S086 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S085 S087 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S086 S088 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S087 S089 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S088 S090 · impliesBefore parent r=1;subset x=3;left product r*x=3;right product x*r=3 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S089 S091 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S090 S092 · impliesBefore parent r=1;subset x=5;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S091 S093 · impliesBefore parent r=1;subset x=6;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S092 S094 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S093 S095 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S094 S096 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S095 S097 · impliesBefore parent r=2;subset x=2;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S096 S098 · impliesBefore parent r=2;subset x=3;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S097 S099 · impliesBefore parent r=2;subset x=4;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S098 S100 · impliesBefore parent r=2;subset x=5;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S099 S101 · impliesBefore parent r=2;subset x=6;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S100 S102 · impliesBefore parent r=2;subset x=7;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S101 S103 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S102 S104 · impliesBefore parent r=3;subset x=1;left product r*x=3;right product x*r=3 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S103 S105 · impliesBefore parent r=3;subset x=2;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S104 S106 · impliesBefore parent r=3;subset x=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S105 S107 · impliesBefore parent r=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S106 S108 · impliesBefore parent r=3;subset x=5;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S107 S109 · impliesBefore parent r=3;subset x=6;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S108 S110 · impliesBefore parent r=3;subset x=7;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S109 S111 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S110 S112 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S111 S113 · impliesBefore parent r=4;subset x=2;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S112 S114 · impliesBefore parent r=4;subset x=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S113 S115 · impliesBefore parent r=4;subset x=4;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S114 S116 · impliesBefore parent r=4;subset x=5;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S115 S117 · impliesBefore parent r=4;subset x=6;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S116 S118 · impliesBefore parent r=4;subset x=7;left product r*x=3;right product x*r=3 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S117 S119 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S118 S120 · impliesBefore parent r=5;subset x=1;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S119 S121 · impliesBefore parent r=5;subset x=2;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S120 S122 · impliesBefore parent r=5;subset x=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S121 S123 · impliesBefore parent r=5;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S122 S124 · impliesBefore parent r=5;subset x=5;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S123 S125 · impliesBefore parent r=5;subset x=6;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S124 S126 · impliesBefore parent r=5;subset x=7;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S125 S127 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S126 S128 · impliesBefore parent r=6;subset x=1;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S127 S129 · impliesBefore parent r=6;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S128 S130 · impliesBefore parent r=6;subset x=3;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S129 S131 · impliesBefore parent r=6;subset x=4;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S130 S132 · impliesBefore parent r=6;subset x=5;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S131 S133 · impliesBefore parent r=6;subset x=6;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S132 S134 · impliesBefore parent r=6;subset x=7;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S133 S135 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S134 S136 · impliesBefore 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S135 S137 · impliesBefore parent r=7;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S136 S138 · impliesBefore parent r=7;subset x=3;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S137 S139 · impliesBefore parent r=7;subset x=4;left product r*x=3;right product x*r=3 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S138 S140 · impliesBefore parent r=7;subset x=5;left product r*x=5;right product x*r=5 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S139 S141 · impliesBefore parent r=7;subset x=6;left product r*x=6;right product x*r=6 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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S140 S142 · impliesBefore parent r=7;subset x=7;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 dependenciesRule: ideal_both_absorptions_all_parent_elements · Uses: G001, S141 S143 · impliesBefore missing left [r,x,rx]=[];missing right [x,r,xr]=[] After left absorbing=Yes;right absorbing=Yes;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 dependenciesRule: ideal_absorption_not_additive_subgroup · Uses: G001, S142 S144 · impliesBefore additive subgroup=Yes;left absorbing=Yes;right absorbing=Yes After two-sided ideal=Yes 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 dependenciesRule: ideal_three_complete_obligations_conjunction · Uses: G001, S143 S145 · impliesBefore actual zero=0;subset=[0, 1, 2, 3, 4, 5, 6, 7];full carrier=[0, 1, 2, 3, 4, 5, 6, 7] After zero ideal=No;whole ring=Yes;supplied subset qualified=Yes Conditions: none {0} means the actual zero singleton,not the empty set.From r0=r(0+0)=r0+r0,additive cancellation gives r0=0;likewise0r=0,so the zero ideal is two-sided.The whole ring qualifies by closure.Still audit the supplied subset rather than assuming it is one of these two. Step identifiers and dependenciesRule: zero_and_whole_ideals_not_empty_subset · Uses: G001, S144 ## Reverse checks - S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S143 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S139 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S137 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S131 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S127 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S125 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 3, 4, 5, 6, 7], No, Yes, Yes, zero_and_whole_are_always_two_sided_ideals) 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: 5ae346896a607213dafcd127ff8852d2af3455ca99c9cdc75f8638f19459b792 - candidate record SHA-256: aa388b84a2940685fb133e9f4b67bdb8a88dee4dbc5bdb55853033b9e381c3e8 - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r02-finite-two-sided-ideal-0093.json)