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学术报告九十四:Lia Vas — Strongly Semihereditary Rings and Rings with Dimension

鸿博体育(集团)有限公司:2022-09-26 作者: 点击数:

报告鸿博体育(集团)有限公司2022年9月29日(星期四)19:30-21:30

报告地点Zoom:953 4167 4893, 密码:212121

Lia Vas 教授

工作单位:University of the Sciences in Philadelphia

举办单位鸿博体育(集团)有限公司

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.


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