= Torsion module
{title2=$T(M)=M$}
{wiki}
A torsion module over an <integral domain> $R$ is a <module> in which every element is a <torsion element of a module>. Each element may require a different nonzero annihilating scalar. The <module> $\mathbb Z/6\mathbb Z$ is torsion over $\mathbb Z$, and the <polynomial module from a linear operator> $\mathbb Q[t]/(t^2)$ is torsion over $\mathbb Q[t]$.
If the <torsion module> is a <finitely generated module>, one common nonzero scalar kills the whole <module>: multiply annihilating scalars for a finite generating set. Without finite generation this can fail. The <quotient module> $\mathbb Q/\mathbb Z$ is torsion, since the class of $a/b$ is killed by $b$, but no nonzero <integer> $n$ kills it all: the class of $1/(2|n|)$ survives multiplication by $n$. See https://kconrad.math.uconn.edu/blurbs/linmultialg/modulesoverPID.pdf[Keith Conrad's Modules over a PID, exercise 6.5(c)].
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