Dual Rickart Modules

Abstract

Rickart property for modules has been studied recently. In this article, we introduce and study the notion of dual Rickart modules. A number of characterizations of dual Rickart modules are provided. It is shown that the class of rings R for which every right R-module is dual Rickart is precisely that of semisimple artinian rings, the class of rings R for which every finitely generated free R-module is dual Rickart is exactly that of von Neumann regular rings, while the class of rings R for which every injective R-module is dual Rickart is precisely that of right hereditary ones. We show that the endomorphism ring of a dual Rickart module is always left Rickart and obtain conditions for the converse to hold true. We prove that a dual Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is a dual Baer module. A structure theorem for a finitely generated dual Rickart module over a commutative noetherian ring is provided. It is shown that, while a direct summand of a dual Rickart module inherits the property, direct sums of dual Rickart modules do not. We introduce the notion of relative dual Rickart property and show that if M i is M j -projective for all i > j ∈ ℐ = {1, 2,…, n} then is a dual Rickart module if and only if M i is M j -d-Rickart for all i, j ∈ ℐ. Other instances of when a direct sum of dual Rickart modules is dual Rickart, are included. Examples which delineate the concepts and results are provided.