On classes of C3 and D3 modules

Authors : Abyzov Adel NAİLEVİCH, Truong Cong QUYNH, Tran Hoai Ngoc NHAN

Pages : 317-329

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Publication Date : 2018-04-01

Article Type : Research Paper

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 mod ules 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 $J^2insert ignore into journalissuearticles values(R);=0$ 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.
Keywords : mathcal A C3 module, mathcal A D3 module, V module