Let M be a left R-module. Recall that the annihilator Ann(x) of an element x in M is the subset {r in R | rx=0} of R, and ideal of R. By the first isomorphism theorem, Ann(x) is an R-submodule of R (i.e., a left ideal), and the quotient abelian group R/Ann(x) is isomorphic as a left R-module to the cyclic submodule Rx of M generated by x.

A nonzero element x in M is a torsion element if Ann(x) is not the zero submodule, i.e., if there exists a nonzero r in R such that rx=0 in M. The module M is a torsion module if every nonzero element of M is a torsion element. For example, any finite abelian group is a torsion Z-module.

If M is a module over an integral domain R, then the set of torsion elements of M (together with 0) forms a submodule, called the torsion submodule of M. Here is the proof: if x and y are torsion, then there are nonzero ring elements r and s such that rx = 0 and sy = 0. Then rs is not zero, and rs(x+y) = 0, so x+y is torsion. If x is torsion, then rx = 0 for som nonzero r. Then for any s in R, s(rx) = 0, so r(sx) = 0, and sx is torsion. 0 is torsion since (1)(0) = 0. (If R is not an integral domain, then the torsion elements of R, considered as a module over itself, are the zero divisors. The set of zero divisors of R is not closed under addition.)