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, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 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] 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, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 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] 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).
Step-by-step derivation
S001 · implies
Before
same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 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 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, 5, 4, 7, 2, 1, 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 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
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 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 dependencies
Rule: ideal_additive_subgroup_nonempty_zero_and_closure · Uses: G001, S007
S009 · 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, S008
S010 · 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, S009
S011 · 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, S010
S012 · 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, S011
S013 · 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, S012
S014 · 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, S013
S015 · 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, S014
S016 · 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, S015
S017 · 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, S016
S018 · implies
Before
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 dependencies
Rule: ideal_three_complete_obligations_conjunction · Uses: G001, S017
S019 · implies
Before
actual zero=0;subset=[0];full carrier=[0, 1, 2, 3, 4, 5, 6, 7]
After
zero ideal=Yes;whole ring=No;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 dependencies
Rule: zero_and_whole_ideals_not_empty_subset · Uses: G001, S018