On rings all of whose modules are retractable. (English) Zbl 1203.16006

Ecevit, Şule; Koşan, Muhammet Tamer

Arch. Math., Brno 45, No. 1, 71-74 (2009).

Summary: Let \(R\) be a ring. A right \(R\)-module \(M\) is said to be retractable if \(\operatorname{Hom}_R(M,N)\neq 0\) 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 prove 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.

MSC:

16D80	Other classes of modules and ideals in associative algebras
16S36	Ordinary and skew polynomial rings and semigroup rings
16D70	Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D90	Module categories in associative algebras

Keywords:

retractable modules; Morita invariants