Torsion submodule

Remarks

The set $T(M)$ is a submodule: if $r m = 0$ and $s n = 0$ with $r, s$ non-zero-divisors, then $rs$ is a non-zero-divisor and $rs(m + n) = 0$, and $r(am) = a(rm) = 0$ for $a \in R$. Example: for $R = \Z$ and $M = \Z \oplus \Z/6\Z$, one has $T(M) = 0 \oplus \Z/6\Z$, since $6 (k, c) = (6k, 0)$ is zero exactly when $k = 0$. Non-example: the element $(1, 0)$ of this $M$ is not torsion, because $n (1, 0) = (n, 0) \ne 0$ for every nonzero integer $n$. When $R$ is an integral domain, the non-zero-divisors are the nonzero elements, and $M / T(M)$ is torsion-free. For $R = \Z$ the torsion submodule is the torsion subgroup.