Simple continuous modules

Year 2019, Volume: 48 Issue: 5, 1336 - 1344, 08.10.2019

Yongduo Wang

Abstract

A module $M$ is called a simple continuous module if it satisfies the conditions $(min-C_{1})$ and $(min-C_{2})$. A module $M$ is called singular simple-direct-injective if for any singular simple submodules $A$, $B$ of $M$ with $A\cong B\mid M$, then $A\mid M$. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple continuous modules and singular simple-direct-injective modules. For instance, it is shown that a ring $R$ is a right $V$-ring if and only if every right $R$-module is a simple continuous modules, and that a regular ring $R$ is a right $GV$-ring if and only if every cyclic right $R$-module is a singular simple-direct-injective module.

Keywords

simple continuous module, singular simple-direct-injective module, $V$-ring, $GV$-ring

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