On dual Rickart modules and weak dual Rickart modules

Let R be a ring. A right R-module M is called d-Rickart if for every endomorphism phi of M, phi(M) is a direct summand of M and it is called wd-Rickart if for every nonzero endomorphism phi of M, phi(M) contains a nonzero direct summand of M. We begin with some basic properties of (w)d-Rickart modules. Then we study direct sums of (w)d-Rickart modules and the class of rings for which every finitely generated module is (w)d-Rickart. We conclude by some structure results.