Dimension and Torsion Theories for a Class of Baer *-Rings

Abstract

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ 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 ${\mathcal C}$. First, we show that a finitely generated module over a ring from the class ${\mathcal C}$ 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 ${\mathcal C}$ and prove that this dimension has all the nice properties of the dimension studied in [W. L\"{u}ck, Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$-Betti numbers I: Foundations, 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 ${\mathcal C},$ we can embed it in a canonical way into a regular ring $Q$ also in ${\mathcal C}.$ We show that $K_0(R)$ is isomorphic to $K_0(Q)$ by producing an explicit isomorphism and its inverse of monoids Proj$ (P)\to$ Proj$ (Q)$ that extends to the isomorphism of $K_0(R)$ and $K_0(Q)$.