Title: Characterizations and direct sums of unit endoregular modules
Speaker: Gangyong Lee (Sungkyunkwan University, Seoul, South Korea)
Abstract: The notion of unit-regular rings was introduced by Ehrlich in 1968. Since then unit-regular rings have attracted wide interest and have been related to many other rings such as regular rings, morphic rings, clean rings, directly finite rings, rings having stable range 1, and rings having the internal cancellation property. In 2013, Lee, Rizvi, and Roman
systematically investigated modules whose endomorphism rings are (von Neumann) regular. Recently, Lee, Roman, and Zhang considered the case in which the endomorphism ring of a module is a division ring. In between
these two cases lies the study of modules whose endomorphism rings are
unit-regular. Our first paper related to unit-endoregular modules has
recently been accepted for publication by Communications in Algebra.
That has now led us to new further research.
In this talk, we introduce the notion of a unit endoregular module as a
module theoretic analogue of a unit-regular ring. We provide different
characterizations of a unit endoregular module from the results of the
first paper. A right $R$-module $M$ is called unit endoregular if its
endomorphism ring is unit-regular. We also discuss when the direct sums
of two or more unit-endoregular modules are unit-endoregular (this is a
joint work with Xiaoxiang Zhang, Southeast University, Nanjing, China).
