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, 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).
Abstract. Motivated by the study of traces on graph C * -algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph C * -algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as canonical since they are uniquely determined by their values on the vertices.A desirable property of a C-valued trace on a C * -algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra L K (E) with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, K-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, C-valued (algebra) traces on L C (E) of a countable graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph C * -algebra C * (E).With the direct finite condition (i.e xy = 1 implies yx = 1) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of "local" Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.
We prove that every perfect torsion theory for a ring R is differential (in the sense of [P.E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1-8]). In this case, we construct the extension of a derivation of a right R-module M to a derivation of the module of quotients of M. Then, we prove that the Lambek and Goldie torsion theories for any R are differential.
We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential.We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.
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