DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

The submodule $I M$ generated by products of elements of an ideal $I$ with elements of a module $M$

Definition

Let $R$ be a commutative ring, let $I ⊴ R$ be an ideal (Left, right and two-sided ideals), and let $M$ be an $R$ -module (Unital left and right modules over a ring; unqualified module means left module). The product of $I$ and $M$ is

$I M : = {\sum_{k = 1}^{t} i_{k} m_{k} : t \in N, i_{k} \in I, m_{k} \in M},$

where the case $t = 0$ is the empty sum $0$ . It is the submodule of $M$ generated by the products $i m$ : sums and negatives remain of the displayed form, and $r (i m) = (r i) m$ with $r i \in I$ .

Thus $0 M = 0$ and $R M = M$ ; the latter follows from $m = 1_{R} m$ .

Depends on

Used by

Dependency tree · two levels

5 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

Stacks Project, Section 10.12: Tensor products (standard reference, not scraped)

The submodule IMIM generated by products of elements of an ideal II with elements of a module MM