Time

Nov 6 2009 - 4:45pm - 5:45 pm

Location

MW 154

Speaker

Gangyong Lee (The Ohio State University)

Seminar Website

http://www.math.ohiou.edu/~algebra/cra/activities.php

Abstract

Let $M$ be a right $R$-module and $S= End_R(M)$. $M_R$ is called
a \emph {Rickart module} if the right annihilator in $M$ of any
single element of $S$ is generated by an idempotent in $S$.

In this talk, we will discuss characterizations of classes of rings
using the Rickart property of modules over them. In particular, the classes of
right semihereditary rings, right hereditary rings, von Neumann regular rings,
$V$-rings and semisimple artinian rings, will be characterized. We will show
that a commutative domain $R$ is a Pr\"{u}fer domain iff the free module
$R^{(2)}$ is a Rickart $R$-module iff  the free module $R^{(3)}$ has
the SIP(Summand Intersection Property). We show that the SIP property
for $R^{(2)}$ in the preceding result is not enough to get that $R$ is
Pr\"{u}fer. We exhibit an example of a domain $R$ for which the free
$R$-module $R^{(2)}$ has the SIP, yet $R$ is not a Pr\"{u}fer domain.
As an application of our results, we provide an alternate proof of
an earlier result of Small by using the theory of Rickart modules:
For any $k\in \mathbb{N}$, $R$ is a right (semi)hereditary ring iff
$\mathbb{M}_k(R)$ is a right (semi)hereditary ring.

(This is a joint work with S. Tariq Rizvi and Cosmin Roman.)