A GENERALIZATION OF CORETRACTABLE MODULES

Abstract

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