Torsion elements, torsion modules, and torsion-free modules

Let \(R\) be a commutative ring and \(M\) an \(R\)-module.

• An element \(m\in M\) is a torsion element if there exists a nonzero \(r\in R\) with \(rm=0\).

• Write \(M_t\coloneqq\theset{m\in M \st rm=0 \text{ for some nonzero } r\in R}\) for the set of torsion elements of \(M\).

• \(M\) is torsion-free if no nonzero element of \(M\) is a torsion element, that is, \(M_t\subseteq\theset{0}\).

• \(M\) is a torsion module if \(M_t=M\).