**S. Tariq Rizvi**

The Ohio State University, U.S.A.

August 18, 2011

**Baer rings: A module theoretic analogue and related notions**:
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 a module, we recently extended these two notions to a general module theoretic setting:
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 $S$. The module $M$ 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, $r_M(\phi)=Ker \phi \leq^\oplus M$ for every $\phi$ in $S$. In this talk we will compar