Abstract
Many 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, 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
C,
we can embed it in a canonical way into a regular ring
Q
also in
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)→
Proj
(Q)
that extends to the isomorphism of
K
0
(R)
and
K
0
(Q)
.