DOIAbstract
AbstractThe notion of Rickart modules was defined recently. It has been shown that a direct sum of Rickart modules is not a Rickart module, in general. In this paper we investigate the question: When are the direct sums of Rickart modules, also Rickart? We show that if Mi is Mj-injective for all i<j∈I={1,2,…,n} then ⊕i=1nMi is a Rickart module if and only if Mi is Mj-Rickart for all i,j∈I. As a consequence we obtain that for a nonsingular extending module M, E(M)⊕M is always a Rickart module. Other characterizations for direct sums to be Rickart under certain assumptions are provided. We also investigate when certain classes of free modules over a ring R, are Rickart. It is shown that every finitely generated free R-module is Rickart precis...
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