  The study of modules over a finite von Neumann algebra ${\mathcal A}$ can be
advanced by the use of torsion theories. In this work, some torsion theories
for ${\mathcal A}$ are presented, compared and studied. In particular, we prove
that the torsion theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$ (in which a
module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie
torsion theories for ${\mathcal A}$.
  Using torsion theories, we describe the injective envelope of a finitely
generated projective ${\mathcal A}$-module and the inverse of the isomorphism
$K_0({\mathcal A})\to K_0({\mathcal U}),$ where ${\mathcal U}$ is the algebra
of affiliated operators of ${\mathcal A}.$ Then, the formula for computing the
capacity of a finitely generated module is obtained. Lastly, we study the
behavior of the torsion and torsion-free classes when passing from a subalgebra
${\mathcal B}$ of a finite von Neumann algebra ${\mathcal A}$ to ${\mathcal
A}$. With these results, we prove that the capacity is invariant under the
induction of a ${\mathcal B}$-module.
