Consider $A$ a ring, $M$ an A-module, and $Tor(M)$ being the set of torsion elements of M (that is, the set of $m \in M$ for which $am=0$ for some $a \in A\backslash \{0\}$ )
Show that $Tor(M)$ need not be a submodule of $M$ if $A$ is not an integral domain.
Apparently I should be considering A as a module over itself, but I don't know what that means. Would a simple counterexample suffice? And if so, is there one?
A ring is a module over itself using the standard multiplication in the ring. Consider the ring $R$ of functions $\mathbb{R}\to\mathbb{R}$. Then in $R$ a function $f$ is torsion iff it has a zero. But the sum of a function $f$ with a zero at $0$ and a function $g$ with a zero at $1$ need not have any zero, hence need not be torsion.