PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

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Torsion elements and $p$ -primary elements form submodules over a domain

Statement

Let $R$ be an integral domain and $M$ an $R$ -module. The torsion subset $Tor (M)$ is a submodule. For every irreducible $p \in R$ , the $p$ -primary component $M [p^{\infty}]$ is a submodule of $Tor (M)$ .

Facts & Assumptions

Given: An integral domain $R$ , an $R$ -module $M$ , the $p$ -primary definition of The $p$ -primary component of a module over a domain, and the submodule test of Submodule of a module.

[F1]

If $R$ is an integral domain, an element $m \in M$ is a torsion element when $r m = 0_{M}$ for some nonzero $r \in R$ (Annihilators, torsion elements and the torsion subset of a module).

Proof

technique · direct

1.1F1algebra

The zero element is torsion. If $a m = 0$ and $b n = 0$ with $a, b \neq 0$ , then $a b \neq 0$ and $a b (m + n) = b (a m) + a (b n) = 0$ ; if $c \in R$ , then $a (c m) = c (a m) = 0$ . Closure under negatives is the scalar case $c = - 1$ , so $Tor (M)$ is a submodule.

2.1givenalgebra∎

The zero element lies in $M [p^{\infty}]$ . If $p^{k} m = 0$ and $p^{ℓ} n = 0$ with $k, ℓ \geq 1$ , then $p^{\max (k, ℓ)} (m + n) = 0$ , and $p^{k} (c m) = 0$ for every $c \in R$ . Thus $M [p^{\infty}]$ is a submodule contained in $Tor (M)$ . The proof includes $k = 1$ , the zero module, and replacing $p$ by an associate.

Depends on

Dependency tree · 2 levels

Sources

K. Conrad, Modules over a PID, Section 4

Torsion elements and pp-primary elements form submodules over a domain