Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)
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A nonzero module over a Noetherian ring has an associated prime

Statement

Assume Dependent Choice.

Let R be a Noetherian commutative ring and let M be a nonzero left R-module. Then Ass⁡R(M) is nonempty.

Facts & Assumptions

Given: Dependent Choice, a Noetherian commutative ring R and a nonzero left R-module M (The axiom of dependent choice: a relation in which every element is related to something admits an N-indexed chain).

[L1]

A prime ideal belongs to Ass⁡R(M) exactly when it is the annihilator of some element of M (Associated primes of a module).

[L2]

Under Dependent Choice, some nonzero element of M has annihilator maximal among annihilators of nonzero elements (A nonzero module over a Noetherian ring has a maximal element annihilator).

[L3]

An annihilator maximal among annihilators of nonzero elements is prime (A maximal element annihilator is prime).

Proof

technique · direct
1.1L2L3choose

By [L2], choose m∈M with m≠0 such that Ann⁡R(m) is maximal among the annihilators of nonzero elements of M. Then [L3] shows that Ann⁡R(m) is prime.

2.1L1step 1.1

By [L1], the prime ideal Ann⁡R(m) belongs to Ass⁡R(M).

3.1step 2.1∎

Therefore Ass⁡R(M) is nonempty.

Depends on

Dependency tree · 2 levels

Sources