On dual Rickart modules and weak dual Rickart modules

Abstract

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