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Keywords:
retractable module; Morita invariant property
retractable module; Morita invariant property
Summary:
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.
References:
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[2] Khuri, S. M.: Endomorphism rings of nonsingular modules. Ann. Sci. Math. Québec 4 (2) (1980), 145–152. MR 0599052 | Zbl 0451.16021
[3] Khuri, S. M.: The endomorphism rings of a non-singular retractable module. East-West J. Math. 2 (2) (2000), 161–170. MR 1825452
[4] Rizvi, S. T., Roman, C. S.: Baer and quasi-Baer Modules. Comm. Algebra 32 (1) (2004), 103–123. DOI 10.1081/AGB-120027854 | MR 2036224 | Zbl 1072.16007
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