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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Second isomorphism theorem for rings: $S / (S \cap I) ≅ (S + I) / I$

Statement

Second isomorphism theorem for rings: $S / (S \cap I) ≅ (S + I) / I$ .

If $S$ is a unital subring of $R$ and $I ⊴ R$ , then this is an isomorphism of unital rings.

Facts & Assumptions

Given: A unital subring $S \subseteq R$ and a two-sided ideal $I ⊴ R$ .

[L1]

$S + I$ is a subring, $I ⊴ S + I$ , and $S \cap I ⊴ S$ (If $S$ is a subring and $I$ is an ideal of $R$ , then $S + I$ is a subring, $I$ is an ideal of $S + I$ , and $S \cap I$ is an ideal of $S$ ).

[L2]

The first ring isomorphism theorem identifies a ring modulo a kernel with its image (First isomorphism theorem for rings: $R / \ker f ≅ im f$ ).

[L3]

The canonical quotient map is a surjective ring homomorphism (The canonical projection $R \to R / I$ is a surjective ring homomorphism with kernel $I$ ).

Proof

technique · direct

1.1

Restrict the quotient map $S + I \to (S + I) / I$ to $ϕ : S \to (S + I) / I$ , $ϕ (s) = s + I$ .

L1L2L3givenconstruct

2.1

Its kernel is $S \cap I$ , and every $(s + i) + I$ equals $s + I$ , so its image is all of $(S + I) / I$ .

step 1.1L1L2L3givenalgebra

3.1

The kernel and image computation gives $S / (S \cap I) ≅ (S + I) / I$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Judson, Abstract Algebra: Theory and Applications, Ring Homomorphisms and Ideals (standard reference, not scraped)

Second isomorphism theorem for rings: S/(S∩I)≅(S+I)/IS/(S∩I)≅(S+I)/I