# Quotienting Z/4Z by its even ideal [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-ring/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-finite-quotient-ring-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables. A complete four-element parent ring and two-element quotient; no claim that every proper-ideal quotient is a field. ## Givens and symbols - calc - G1 — R=Z/4Z, I={0,2}, with addition and multiplication modulo 4. - R — parent ring modulo 4 - I — even ideal - E,O — complete additive cosets ## Rules and sources - calc · prerequisite — Assume integer congruence and ring laws; an ideal is an additive subgroup absorbing multiplication from both sides. - mko_definition · cited — Define a equivalent to b when a−b belongs to I. The classes are additive cosets, with [a]+[b]=[a+b] and [a][b]=[ab]. Check representative independence completely. Because this project requires 1 distinct from 0 in a unital ring, the degenerate zero ring from I=R is excluded. · mko-finite-quotient-ring ## Main derivation S01 · impliesBefore R=\{0,1,2,3\} After \begin{array}{c|cccc}+&0&1&2&3\\\hline 0&0&1&2&3\\1&1&2&3&0\\2&2&3&0&1\\3&3&0&1&2\\\end{array}\qquad\begin{array}{c|cccc}*&0&1&2&3\\\hline 0&0&0&0&0\\1&0&1&2&3\\2&0&2&0&2\\3&0&3&2&1\\\end{array} Conditions: none Rule: calc · Uses: G1 Leaf basis: calc All 32 parent-ring entries are listed. Integer ring laws descend because congruence modulo 4 respects addition and multiplication. S02 · impliesBefore I=\{0,2\} After 0+0=0,\ 0+2=2,\ 2+0=2,\ 2+2=0;\quad -0=0,-2=2 Conditions: none Rule: calc · Uses: G1, S01 Leaf basis: calc Every sum and inverse in I is listed, proving it is an additive subgroup. S03 · impliesBefore r\in R,\quad i\in I After r0=0r=0;\quad(0,1,2,3)2=(0,2,0,2)=2(0,1,2,3) Conditions: none Rule: calc · Uses: G1, S01, S02 Leaf basis: calc Every left and right product lies in I. Since 1 is outside I, this is a proper two-sided ideal. S04 · impliesBefore a\sim b\iff a-b\in I After E=0+I=2+I=\{0,2\},\quad O=1+I=3+I=\{1,3\} Conditions: none Rule: calc · Uses: G1, S03 Leaf basis: calc Ideal congruence is parity equality; all cosets are listed and deduplicated. S05 · impliesBefore a'=a+i,\quad b'=b+j,\quad i,j\in I After (a'+b')-(a+b)=i+j\in I Conditions: none Rule: calc · Uses: G1, S02, S04 Leaf basis: calc Changing either representative changes the sum by an ideal element, proving well-defined addition. S06 · impliesBefore a'=a+i,\quad b'=b+j After a'b'-ab=aj+ib+ij\in I Conditions: none Rule: calc · Uses: G1, S02, S03, S04 Leaf basis: calc Two-sided absorption and additive closure put all three terms in I, proving representative-independent multiplication. S07 · impliesBefore R/I=\{E,O\} After \begin{array}{c|cc}+&E&O\\\hline E&E&O\\O&O&E\\\end{array}\qquad\begin{array}{c|cc}*&E&O\\\hline E&E&E\\O&E&O\\\end{array} Conditions: none Rule: calc · Uses: G1, S01, S04, S05, S06 Leaf basis: calc Compute every pair of classes from the parent tables; the preceding proofs cover all representative substitutions. S08 · impliesBefore 0+I=E,\quad1+I=O After E\ne O,\quad R/I\text{ is a unital quotient ring} Conditions: none Rule: mko_definition · Uses: G1, S03, S05, S06, S07 Leaf basis: mko_definition Because 1 is not in I, zero and one remain distinct; ring identities descend through well-defined quotient operations. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, S03, S05, S06, S07 — If I=R only one class remains and 1=0, violating this project's unital-ring convention. - S07 · premise_audit · G1, S01, S04, S05, S06 — Each quotient table has four entries; E is the zero class and O the one class. - S06 · premise_audit · G1, S02, S03, S04 — The order aj,ib matters even in a noncommutative parent; commutativity here does not remove the ideal obligation. - S05 · premise_audit · G1, S02, S04 — i and j range over all of I, not just a preferred representative pair. - S04 · premise_audit · G1, S03 — E and O are disjoint and cover R; quotient elements are not single representatives 0 or 1. - S03 · premise_audit · G1, S01, S02 — Both sides are checked; cardinality alone cannot certify an ideal. - S02 · premise_audit · G1, S01 — The inverse of 2 remains in I; nonemptiness and additive closure are both checked. - S01 · premise_audit · G1 — Additive zero 0 and multiplicative one 1 are distinct; quotient tables do not replace the parent tables. ## Error controls and negative controls Ideal firstCheck additive subgroup and two-sided absorption before quotient multiplication. Representative changesAll three terms of a'b'-ab must lie in I; four preferred-representative products alone are insufficient. - R/I={0,1} as a subset of the parent. → reject — Its carrier is the two coset sets E,O; 0,1 are merely chosen representatives. - I=R also satisfies this project's 1 distinct from 0 convention. → reject — Then 0+I=1+I=R. - Every proper-ideal quotient is a field. → reject — The zero ideal of Z/4Z is proper, but its quotient retains nonzero 2 with 2^2=0. ## Result and limitations I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables. ## Verification and source binding 1 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: c45f9867151cba8da988134e5c080121c96a4971332e3fd16a292f9facd2df8b - original source SHA-256: c45f9867151cba8da988134e5c080121c96a4971332e3fd16a292f9facd2df8b - Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW [Same-source full JSON (formulas, ASTs, controls and revision log)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-finite-quotient-ring-01.json)