On dual Rickart modules and weak dual Rickart modules

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.