`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 compare and contrast
the two notions and present their properties. Endomorphism ringsof these modules 
and their direct sums will be discussed. We will
present some recent developments in this theory including a dual notion.
(This is a joint work with Gangyong Lee and Cosmin Roman.)