Date & Time: August 19, 2011; 16.45

Venue: Ramanujan Hall

Title: Theory of Baer Modules- Some Recent Developments

Speaker: S. Tariq Rizvi, Department of Mathematics, The Ohio State University Lima, Ohio, U. S. A.

Abstract:Kaplansky introduced the notion of a Baer ring in 1955 which has close links to $C^*$-algebras and von Neumann algebras. Maeda and Hattori generalized this notion to that of a Rickart Ring in 1960. A ring is called Baer (right Rickart) if the right annihilator of any subset (single element) of $R$ is generated by an idempotent of $R$.

Using the endomorphism ring of the module, we extended these two notions to a general module theoretic setting in 2004 and 2010 respectively: Let $R$ be any ring, $M$ be an $R$-module and $S =End_R(M)$. $M$ is said to be a {\it Baer module} if the right annihilator in $M$ of any subset of $S$ is generated by an idempotent of $S$. Equivalently, the left annihilator in $S$ of any submodule of $M$ is generated by an idempotent of $S$. $M$ is called a {\it Rickart module} if the right annihilator in $M$ of any single element of $S$ is generated by an idempotent of $S$, equivalently, $r_M(\varphi)=Ker \varphi \leq^\oplus M$ for every $\varphi \in S$. Among other properties of these two notions we show that while each is inherited by direct summands of modules, the properties do not carry over to direct sums of such modules. We will discuss when such direct sums also inherit these notions and present recent developments in this theory.