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

1.1

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

givenL1

1.2

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 $J (R / I)$ .

givenL2algebra

2.1

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

step 1.1step 1.2L3

3.1

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.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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