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

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.

The sum and product of two-sided ideals are two-sided ideals

Statement

The sum and product of two-sided ideals are two-sided ideals.

Facts & Assumptions

Given: Two-sided ideals $I, J ⊴ R$ .

[L1]

$I + J$ and $I J$ are the indicated elementwise and finite-sum sets (The sum $I + J$ and product $I J$ of two-sided ideals).

[L2]

The ideal criterion is subtraction closure plus two-sided absorption (Ideal criteria and intersections of ideals).

[L3]

Ring multiplication distributes over finite sums (In any ring $0 \cdot a = a \cdot 0 = 0$ , $(- a) b = a (- b) = - (a b)$ , $(- a) (- b) = a b$ , $(- 1) a = - a$ and $a (b - c) = a b - a c$ ).

Proof

technique · direct

1.1

Subtraction closure of $I + J$ and of $I J$ follows by subtracting representatives and concatenating finite sums.

L1L2L3givenalgebra

2.1

Multiplying a representative on either side keeps it in $I + J$ , and distributivity keeps each summand of a product in $I J$ .

step 1.1L1L2L3givenalgebra

3.1

The closure established in step 2.1 proves both claims.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)