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 compare and contrast the two notions and present their properties. Endomorphism rings of 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).