Algebras and Representation Theory

, Volume 15, Issue 6, pp 1049–1079

Strongly Semihereditary Rings and Rings with Dimension

Article

Abstract

The existence of a well-behaved dimension of a finite von Neumann algebra (see Lück, J Reine Angew Math 495:135–162, 1998) has lead to the study of such a dimension of finite Baer *-rings (see Vaš, J Algebra 289(2):614–639, 2005) that satisfy certain *-ring axioms (used in Berberian, 1972). This dimension is closely related to the equivalence relation $${\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$$ on projections defined by $$p{\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}} q$$ iff p = xx* and q = x*x for some x. However, the equivalence $${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}}$$ on projections (or, in general, idempotents) defined by $$p{\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}} q$$ iff p = xy and q = yx for some x and y, can also be relevant. There were attempts to unify the two approaches (see Berberian, preprint, 1988)). In this work, our agenda is three-fold: (1) We study assumptions on a ring with involution that guarantee the existence of a well-behaved dimension defined for any general equivalence relation on projections ~. (2) By interpreting ~ as $${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}},$$ we prove the existence of a well-behaved dimension of strongly semihereditary *-rings with positive definite involution. This class is wider than the class of finite Baer *-rings with dimension considered in the past: it includes some non Rickart *-rings. Moreover, none of the *-ring axioms from Berberian (1972) and Vaš (J Algebra 289(2):614–639, 2005) are assumed. (3) As the first corollary of (2), we obtain dimension of noetherian Leavitt path algebras over positive definite fields. Secondly, we obtain dimension of a Baer *-ring R satisfying the first seven axioms from Vaš (J Algebra 289(2):614–639, 2005) (in particular, dimension of finite AW*-algebras). Assuming the eight axiom as well, R has dimension for $${\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$$ also and the two dimensions coincide. While establishing (2), we obtain some additional results for a right strongly semihereditary ring R: we prove that every finitely generated R-module M splits as a direct sum of a finitely generated projective module and a singular module; we describe right strongly semihereditary rings in terms of relations between their maximal and total rings of quotients; and we characterize extending Leavitt path algebras over finite graphs.

Keywords

Dimension Rings of quotients Semihereditary Involution Baer Regular

Mathematics Subject Classification (2010)

16W99 16S99 16S90 16W10

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