The Set of Torsion Elements of a Module

Let R be a commutative ring and M an R-module. The purpose of this paper is to investigate the set T(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} of torsion elements of M. Of particular interest is the question of when T(M) is a submodule of M. We show that if M ≠ T(M), then T(M) is a union of prime submodules of M.