Torsion submodule
= Torsion submodule
{title2=$T(M)$}
For a <module> $M$ over an <integral domain> $R$, its torsion submodule is
$$
T(M)=\{m\in M:rm=0\text{ for some }0\ne r\in R\}.
$$
This is a <submodule>. If $rm=0$ and $sn=0$ with $r,s\ne0$, then $rs(m+n)=0$ and $rs\ne0$; also $r(am)=a(rm)=0$ for every $a\in R$. The <integral domain> hypothesis supplies both the nonzero product and commutativity. These are the conventions of https://kconrad.math.uconn.edu/blurbs/linmultialg/moduleintro.pdf[Keith Conrad's Introductory Notes on Modules, section 4]{ref}.
A mixed example is the <direct sum> $M=\mathbb Z^2\oplus\mathbb Z/6\mathbb Z$. If a nonzero <integer> kills $(a,b,\overline c)$, then $a=b=0$. Conversely, $6$ kills every $(0,0,\overline c)$. Thus
$$
T(M)=\{(0,0,\overline c):\overline c\in\mathbb Z/6\mathbb Z\},\qquad M/T(M)\cong\mathbb Z^2.
$$
The <torsion submodule> isolates the finite-order part while the <quotient module> keeps the independent <integer> directions. In general $M/T(M)$ is a <torsion-free module>: if $r(m+T(M))=0$ with $r\ne0$, some $s\ne0$ kills $rm$, so $sr\ne0$ kills $m$.
The hypothesis matters. In the <ring> $R=\mathbb Z/6\mathbb Z$ viewed as a <module> over itself, the elements $\overline2$ and $\overline3$ are killed by nonzero scalars $\overline3$ and $\overline2$, but their sum $\overline5$ is a <unit> and is killed by no nonzero scalar. Thus using arbitrary nonzero scalars in a <ring> with <zero divisors> need not give a <submodule>. A common broader convention uses <regular elements of a ring> as annihilators; the <integral domain> convention here avoids that ambiguity.