What is torsion? — Mathematics

Definition 3.11 University Mathematics — Year 3 · Chapter 3 — Modules over a Principal Ideal Domain

Let $A$ be a domain and $M$ an $A$ -module. The torsion submodule is

T(M) = \{x \in M : ax = 0 \text{ for some } a \neq 0\}

(a submodule: if $ax = by = 0$ then $ab(x + y) = 0$ , $ab \ne 0$ ). $M$ is torsion-free if $T(M) = 0$ , a torsion module if $T(M) = M$ .

Examples

Example 3.2

The three motivating cases.

$A = K$ a field: modules are vector spaces.
$A = \Z$ : modules are exactly abelian groups ( $nx$ is forced to be $x + \dots + x$ ), submodules are subgroups.
$A = K[X]$ : a module is a $K$ -vector space $V$ together with the $K$ -linear map $u\colon x \mapsto X\cdot x$ — conversely, every pair $(V, u)$ with $u \in \mathcal L(V)$ becomes a $K[X]$ -module by $P \cdot x = P(u)(x)$ . The submodules are precisely the $u$ -stable subspaces.

An ideal of $A$ is exactly a submodule of $A$ ; a quotient ring $A/I$ is an $A$ -module. Unlike vector spaces, modules can have torsion: in $\Z/6\Z$ , the element $\bar 3 \ne 0$ is killed by $2 \neq 0$ .

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Examples

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