Definition 5

In a module $E$ over an integral domain $A$, the torsion submodule of $E$ is the submodule of $E$ consisting of the non-free elements (also called the torsion elements of $E$).

When $E$ is equal to its torsion submodule (that is when every element of $E$ is annihilated by an element $\neq 0$ of $A$) $E$ is called a torsion module. When the torsion submodule of $E$ is reduced to 0 (that is every non-zero element of $E$ is free) $E$ is called (by an abuse of language) a torsion-free module.

Every submodule of a free $A$-module (and in particular every projective $A$-module) is torsion-free. The $\mathbf{Z}$-module $\mathbf{Q}$ is torsion-free.