, Volume 15, Issue 6, pp 1049-1079
Date: 05 Apr 2011

Strongly Semihereditary Rings and Rings with Dimension

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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.

Presented by Kenneth Goodearl.
This paper was completed mostly during the author’s visit to the University of Málaga, partially funded by a “Grant for foreign visiting professors” within the III Research Framework Program of the University of Málaga. The author thanks the host institution for the hospitality and support.