Strongly semihereditary rings and rings with dimension

Abstract

The existence of a well-behaved dimension of a finite von Neumann algebra (see [19]) has lead to the study of such a dimension of finite Baer *-rings (see [26]) that satisfy certain *-ring axioms (used in [9]). This dimension is closely related to the equivalence relation ${\sim }^{\ast }$ on projections defined by $p{\sim }^{\ast }q$ iff $p=x{x}^{\ast }$ and $q=x^{\ast }x$ for some $x.$ However, the equivalence ${\sim }^a$ on projections (or, in general, idempotents) defined by $p{\sim }^{a}q$ iff $p=xy$ and $q=yx$ for some $x$ and $y,$ can also be relevant. There were attempts to unify the two approaches (see [10]). 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 $\sim .$ (2) By interpreting $\sim$ as ${\sim }^a,$ we prove the existence of a well-behaved dimension of strongly semihereditary rings with a 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 [9] and [26] 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 [26] (in particular, dimension of finite $A{W}^{\ast }$-algebras). Assuming the eight axiom as well, $R$ has dimension for ${\sim }^{\ast }$ also and the two dimensions coincide.