报告时间：2022年10月14日（星期四）19:30-21:30

报告地点：Zoom：953 4167 4893, 密码：212121

报 告 人：Lia Vas 教授

工作单位：University of the Sciences in Philadelphia

举办单位：数学学院

报告人简介：Lia Vas，美国费城科技大学教授、博士生导师，2002年获得美国马里兰大学博士学位，2018年为费城科技大学教授，主要研究方向为环论。在J. Algebra., J. Pure. Appl. Algebra, Comm. Algebra等杂志上发表论文30余篇。

报告简介：The existence of a well-behaved dimension of a finite von Neumann algebra (see Luck, J Reine Angew Math 495:135-162, 1998) has lead to the study of such a dimension of finite Baer (*)-rings (see Vas, 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 (sic) on projections defined by p (sic) q iff p = xx(*) and q = x(*)x for some x. However, the equivalence (sic) on projections (or, in general, idempotents) defined by p (sic) 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 similar to. (2) By interpreting similar to as (sic), 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 Vas (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 Vas (J Algebra 289(2):614-639, 2005) (in particular, dimension of finite AW (*)-algebras). Assuming the eight axiom as well, R has dimension for 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.