ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$(R / I) \otimes_{R} (R / J) ≅ R / (I + J)$ for ideals of a commutative ring

Example

Let $I, J$ be ideals of a commutative ring $R$ . There is a canonical isomorphism

$(R / I) \otimes_{R} (R / J) ≅ R / (I + J) .$

This includes $I = 0$ , $J = 0$ , $I = R$ , and $J = R$ .

Facts & Assumptions

Given: A commutative ring $R$ and ideals $I, J \subseteq R$ .

[L1]

For a right $R$ -module $M$ , $M \otimes_{R} R / J ≅ M / J M$ ( $M \otimes_{R} R / I ≅ M / I M$ naturally).

[L2]

$I + J = {i + j : i \in I, j \in J}$ is an ideal (The sum $I + J$ and product $I J$ of two-sided ideals).

[L3]

A module homomorphism induces an isomorphism from its quotient by its kernel to its image (First isomorphism theorem for modules: $M / \ker f ≅ im f$ ).

Verification

technique · direct

1.1givenL1

Apply [L1] with $M = R / I$ to obtain $(R / I) \otimes_{R} (R / J) ≅ (R / I) / J (R / I)$ .

1.2givenL2algebra

The map $q : R / I \to R / (I + J)$ given by $q (r + I) = r + (I + J)$ is well-defined and surjective. Its kernel consists of the classes $i + j + I = j + I$ with $i \in I$ and $j \in J$ , which is exactly $J (R / I)$ .

2.1step 1.1step 1.2L3

By [L3], step 1.2 induces $(R / I) / J (R / I) ≅ R / (I + J)$ ; composing with step 1.1 proves the displayed isomorphism.

3.1step 2.1L2∎

If $I = 0$ or $J = 0$ , the formula reduces to the appropriate tensor-unit isomorphism. If either ideal is $R$ , then both sides are zero. Thus all stated boundary cases are included.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Wenqi Li, Commutative Algebra, Lecture 10 (standard reference, not scraped)

(R/I)⊗R(R/J)≅R/(I+J)(R/I)⊗R​(R/J)≅R/(I+J) for ideals of a commutative ring