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

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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

1.1

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.

F1algebra

2.1

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.

givenalgebra∎

Depends on

Used by

Cited to discharge well-definedness by The p-primary component of a module over a domain.

Dependency tree · two levels

8 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

K. Conrad, Modules over a PID, Section 4 (standard reference, not scraped)

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