Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28
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.
Associated primes of a module
Definition
Let be a commutative ring and let be a left -module. A prime ideal is associated to when
for some element . The set of associated primes of is denoted
Depends on
Used by
- Associated primes of a cyclic quotient are colon primes Corollary
- The zero module has empty support and no associated primes Example
- A zero divisor is contained in an associated prime Lemma
- Associated primes are exactly primes of embedded cyclic residue modules Lemma
- Associated primes lie in the support Lemma
- Associated primes of a localized finite module come from upstairs Lemma
- A nonzero module over a Noetherian ring has an associated prime Theorem
- Primary submodules of finite modules are characterized by a singleton associated-prime set Theorem
Dependency tree · two levels
7 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
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Definition (17.1) (standard reference, not scraped)
- The Stacks Project, Section 10.63: Associated primes (standard reference, not scraped)