The study of modules over a finite von Neumann algebra A can be advanced by the use of torsion theories. In this work, some torsion theories for A are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for A.Using torsion theories, we describe the injective envelope of a finitely generated projective si-module and the inverse of the isomorphism K-0(A) -> K-0(U), where U is the algebra of affiliated operators of 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 B of a finite von Neumann algebra A to A. With these results, we prove that the capacity is invariant under the induction of a B-module.
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