Quotienting Z/4Z by its even ideal

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

R=\{0,1,2,3\}

\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}

I=\{0,2\}

0+0=0,\ 0+2=2,\ 2+0=2,\ 2+2=0;\quad -0=0,-2=2

r\in R,\quad i\in I

r0=0r=0;\quad(0,1,2,3)2=(0,2,0,2)=2(0,1,2,3)

a\sim b\iff a-b\in I

E=0+I=2+I=\{0,2\},\quad O=1+I=3+I=\{1,3\}

a'=a+i,\quad b'=b+j,\quad i,j\in I

(a'+b')-(a+b)=i+j\in I

a'=a+i,\quad b'=b+j

a'b'-ab=aj+ib+ij\in I

R/I=\{E,O\}

\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}

0+I=E,\quad1+I=O

E\ne O,\quad R/I\text{ is a unital quotient ring}

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 first

Check additive subgroup and two-sided absorption before quotient multiplication.

Representative changes

All 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

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