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.

$(S, R)$ -bimodules and commuting left and right scalar actions

Definition

Let $S$ and $R$ be unital rings. An $(S, R)$ -bimodule is an abelian group $N$ that is a left $S$ -module and a right $R$ -module (Unital left and right modules over a ring; unqualified module means left module) such that the two actions commute:

$(s n) r = s (n r)$

for every $s \in S$ , $n \in N$ , and $r \in R$ . It is denoted $_{S} N_{R}$ when the rings need to be displayed.

Every ring $R$ is an $(R, R)$ -bimodule by left and right multiplication. If $R$ is commutative, every left $R$ -module becomes an $(R, R)$ -bimodule by defining $m r : = r m$ .

Depends on

Unital left and right modules over a ring; unqualified module means left module

Used by

Dependency tree · two levels

3 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)

(S,R)(S,R)-bimodules and commuting left and right scalar actions