Direct sums of Rickart modules

Abstract

The 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 M i is M j -injective for all i < j ∈ I = { 1 , 2 , … , n } then ⊕ i = 1 n M i is a Rickart module if and only if M i is M j -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 precisely when R is a right semihereditary ring. As an application, we show that a commutative domain R is Prüfer if and only if the free R-module R ( 2 ) is Rickart. We exhibit an example of a module M for which M ( 2 ) is Rickart but M ( 3 ) is not so. Further, von Neumann regular rings are characterized in terms of Rickart modules. It is shown that the class of rings R for which every finitely cogenerated right R-module is Rickart, is precisely that of right V-rings. Examples which delineate the concepts and the results are provided.