TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal

Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal.

Let $J \leq (R, +)$ . The rule $(r + J) (s + J) = r s + J$ is independent of representatives exactly when $J ⊴ R$ .

Given: A ring $R$ and an additive subgroup $J \leq (R, +)$ .

[L1]

The displayed rule is the proposed quotient multiplication (The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ ).

[L2]

A two-sided ideal is an additive subgroup absorbing multiplication on both sides (Left, right and two-sided ideals).

[L3]

[L4]

[L5]

technique · direct

1.1L1L2L3L4L5givenalgebra

If $J$ is an ideal and $r^{'} = r + i$ , $s^{'} = s + j$ , then $r^{'} s^{'} - r s = r j + i s + i j \in J$ , so the product is well defined.

2.1step 1.1L1L2L3L4L5givenalgebra

Conversely, compare $(r + J) (0 + J)$ with $(r + J) (j + J)$ and the reversed product; [L4] gives $r j, j r \in J$ for all $r, j$ .

3.1step 2.1∎

Thus the rule is well defined exactly when $J$ is a two-sided ideal.

Depends on