Document Type : Original Manuscript


Department of Mathematics, University of Mazandaran, Babolsar, Iran


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.