DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $I + J$ and product $I J$ of two-sided ideals

Definition

The sum $I + J$ and product $I J$ of two-sided ideals.

For two-sided ideals $I, J ⊴ R$ , set

$I + J : = {i + j : i \in I, j \in J}, I J : = {\sum_{k = 1}^{m} i_{k} j_{k} : m \geq 0, i_{k} \in I, j_{k} \in J} .$

The empty sum in the definition of $I J$ is $0$ .

Depends on

Left, right and two-sided ideals

Dependency tree · 2 levels

Sources

Janssen and Lindsey, Rings with Inquiry, Ideals