Let \(R\) be an integral domain and \(M\) an \(R\)-module. The torsion submodule of \(M\) is \[ \tor(M) \coloneqq \theset{m \in M \st rm = 0 \text{ for some nonzero } r \in R}. \]
\(\tor(M)\) is a submodule: if \(rm=0\) and \(sn=0\) with \(r,s\neq0\), then \(rs\neq0\) because \(R\) is a domain, \(rs(m+n)=0\), and \(r(am)=a(rm)=0\) for every \(a\in R\).
What is the torsion submodule $\tor(M)$ of a module $M$?