# On rings all of whose modules are retractable

## Şule Ecevit and Muhammet Tamer Koşan

Address: Department of Mathematics, Gebze Institute of Technology Çayirova Campus, 41400 Gebze-Kocaeli, Turkey

E-mail:

secevit@gyte.edu.tr

mtkosan@gyte.edu.tr

Abstract: Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $\mathbb{T}{Hom}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that
$(1)$ The ring $\prod _{i \in \mathcal{I}} R_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in \mathcal{I}$, where $\mathcal{I}$ is an arbitrary finite set.
$(2)$ If $R[x]$ is a mod-retractable ring then $R$ is a mod-retractable ring.

AMSclassification: primary 16D50; secondary 16D90.

Keywords: retractable module, Morita invariant property.