Modules Whose Endomorphism Rings are Right Rickart
Thoraya Abdelwhab
Department of Mathematics, Northwest Normal University, Lanzhou, Gansu, P.R. China and Faculty of Mathematical Sciences, University of Khartoum, Khartoum, Khartoum, Sudan.
Xiaoyan Yang *
Department of Mathematics, Northwest Normal University, Lanzhou, Gansu, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we study modules whose endomorphism rings are right Rickart (or right p.p.) rings, which we call R-endoRickart modules. We provide some characterizations of R-endoRickart modules. Some classes of rings are characterized in terms of R-endoRickart modules. We prove that an R-endoRickart module with no innite set of nonzero orthogonal idempotents in its endomorphism ring is precisely an endoBaer module. We show that a direct summand of an R-endoRickart modules inherits the property, while a direct sum of R-endoRickart modules does not. Necessary and sucient conditions for a nite direct sum of R-endoRickart modules to be an R-endoRickart module are provided.
Keywords: R-endoRickart module, endoBaer module, Rickart module, right Rickart ring, Baer ring.