AI Summary & Key Takeaways
AI InsightIf all right $R$-modules are $\mathcal{A}$-injective, then $R$ is a serial artinian ring with $J^2(R)=0$ if and only if every $\mathcal{A}$-C3 right $R$-module is quasi-injective, if and only if every $\mathcal{A}$-C3 right $R$-module is C3.
📌 3 Key Takeaways
Abstract
This paper aims to study the notions of $\mathcal{A}$-C3 and $\mathcal{A}$-D3 modules for some class $\mathcal{A}$ of right modules. Several characterizations of these modules are provided and used to describe some well-known classes of rings and modules. For example, a regular right $R$-module $F$ is a $V$-module if and only if every $F$-cyclic module is an $\mathcal{A}$-C3 module, where $\mathcal{A}$ is the class of all simple right $R$-modules. Moreover, let $R$ be a right artinian ring and $\mathcal{A}$, a class of right $R$-modules with a local ring of endomorphisms, containing all simple right $R$-modules and closed under isomorphisms. If all right $R$-modules are $\mathcal{A}$-injective, then $R$ is a serial artinian ring with $J^2(R)=0$ if and only if every $\mathcal{A}$-C3 right $R$-module is quasi-injective, if and only if every $\mathcal{A}$-C3 right $R$-module is C3.