A GENERALIZATION OF CORETRACTABLE MODULES

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $\overline{Z}(M)$ (for short, $\overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $\overline{Z}(M)$, there is a nonzero homomorphism $f:\dfrac{M}{N}\rightarrow M$. We investigate some conditions under which the two concepts coretractable and $\overline{Z}(M)$-coretractable, coincide. For a commutative semiperfect ring $R$, we show that $R$ is $\overline{Z}(R)$ coretractable if and only if $R$ is a Kasch ring. Some examples are provided to illustrate different concepts.