Dimension and torsion theories for a class of Baer *-rings

AbstractMany known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class C of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class C. First, we show that a finitely generated module over a ring from the class C splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from C and prove that this dimension has all the nice properties of the dimension studied in [W. Lück, J. Reine Angew. Math. 495 (1998) 135–162] for finite von Neumann algebras. This dimension defines a torsion theory that we prove to be equal to the Goldie and Lambek torsion theories. Moreover, every finitely generated module splits in this torsion theory. If R is a ring in C, we can embed it in a canonical way into a regular ring Q also in C. We show that K0(R) is isomorphic to K0(Q) by producing an explicit isomorphism and its inverse of monoids Proj(P)→Proj(Q) that extends to the isomorphism of K0(R) and K0(Q)