Descripción
**Hora de inicio:** 12:00. **Hora de fin:** 13:00.
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 a (right) direct summand of $R$. While the notion of a Baer ring is always left-right symmetric,
this does not hold true for the Rickart property of rings.
Using the endomorphism ring of the 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 a direct summand of $M$. Equivalently, the left
annihilator in $S$ of any submodule of $M$ is a direct summand of $S$.
$M$ is called a {\it Rickart module} if the right
annihilator in $M$ of any single element of $S$ is a direct summand of $M$,
equivalently, $r_M(\varphi)=Ker \varphi \leq^\oplus M$ for every $\varphi \in S$.
We will discuss these two notions, present their properties and examples and mention some
recent developments in this new developing theory.