SOME PROPERTIES OF RICKART MODULES

Let R be an arbitrary ring with identity and M a right R-module with S = End( R)(M). Following [8], the module M is called Rickart if for any f( )is an element of S, r(M)(f) = eM for some e(2) = e is an element of S, equivalently, Kerf is a direct summand of M. In this paper, we continue to investigate properties of Rickart modules. For a Rickart module M, we prove that M is S-rigid (resp., S-reduced, S-symmetric, S-semicommutative, S-Armendariz) if and only if its endomorphism ring S is rigid (resp., reduced, symmetric, semicommutative, Armendariz). We also prove that if M[x] is a Rickart module with respect to S[x], then M is Rickart, the converse holds if M is S-Armendariz. Among others it is also shown that M is a Rickart module if and only if every right R-module is M-principally projective.