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, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 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] 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,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.
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, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 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] 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,missing_left [r,x,r*x] rows,missing_right [x,r,x*r] rows,left_absorbing?,right_absorbing?). Both failure lists use parent-r outer order and subset-x inner order.
Step-by-step derivation
S001 · implies
Before
same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 6, 7, 5, 0], [2, 3, 4, 6, 5, 0, 7, 1], [3, 4, 6, 5, 7, 1, 0, 2], [4, 6, 5, 7, 0, 2, 1, 3], [5, 7, 0, 1, 2, 4, 3, 6], [6, 5, 7, 0, 1, 3, 2, 4], [7, 0, 1, 2, 3, 6, 4, 5]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 4, 5, 0, 4, 2, 5], [0, 3, 5, 1, 4, 2, 7, 6], [0, 4, 0, 4, 0, 0, 4, 4], [0, 5, 4, 2, 0, 4, 5, 2], [0, 6, 2, 7, 4, 5, 1, 3], [0, 7, 5, 6, 4, 2, 3, 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 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, 7, 5, 6, 4, 2, 3, 1]
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];same qualified parent zero=0
After
sorted distinct parent labels=[0]
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
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, S005
S007 · 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, S006
S008 · 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, S007
S009 · 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, S008
S010 · 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, S009
S011 · 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, S010
S012 · 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, S011
S013 · 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, S012
S014 · implies
Before
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 dependencies
Rule: ideal_absorption_not_additive_subgroup · Uses: G001, S013