LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

Primary submodules are exactly quotients with nilpotent zero divisors

Statement

Let $R$ be a commutative ring, let $M$ be a left $R$ -module, and let $Q ⊊ M$ be a proper submodule. Then $Q$ is primary if and only if the following classical condition holds:

for every $a \in R$ and every $m \in M$ , if $a m \in Q$ and $m \notin Q$ , then there exists $n \geq 1$ such that $a^{n} M \subseteq Q .$

Facts & Assumptions

Given: A commutative ring $R$ , a left $R$ -module $M$ , and a proper submodule $Q ⊊ M$ .

[L1]

A proper submodule $Q$ is primary exactly when every zero divisor on $M / Q$ acts nilpotently on $M / Q$ (Primary submodules and primary ideals).

[L2]

The quotient module $M / Q$ consists of cosets $m + Q$ (Quotient module $M / N$ with scalar multiplication on additive cosets).

Proof

technique · direct

1.1L1L2givenalgebra

Assume $Q$ is primary, and let $a m \in Q$ with $m \notin Q$ . Then $m + Q \neq 0$ in $M / Q$ by [L2], while $a (m + Q) = a m + Q = 0 + Q$ . Thus $a$ is a zero divisor on $M / Q$ . By [L1], some $n \geq 1$ satisfies $a^{n} (M / Q) = 0$ , which is equivalent to $a^{n} M \subseteq Q$ .

1.2L1L2given

Conversely, assume the displayed classical condition. Let $a$ be a zero divisor on $M / Q$ . Then there exists $m + Q \neq 0$ with $a (m + Q) = 0 + Q$ . By [L2], this means $m \notin Q$ and $a m \in Q$ . The hypothesis gives $n \geq 1$ with $a^{n} M \subseteq Q$ , equivalently $a^{n} (M / Q) = 0$ . Hence every zero divisor on $M / Q$ acts nilpotently, so $Q$ is primary by [L1].

Primary submodules are exactly quotients with nilpotent zero divisors

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources