On classes of C3 and D3 modules

• Published in: arXiv.org 2016-09
• Main Authors: Abyzov Adel Nailevich, Truong, Cong Quynh, Tran Hoai Ngoc Nhan
• Format: Article
• Language: English
• Subjects: Isomorphism; Modules
• ISSN: 2331-8422

The aim of this paper is 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 \(M\) is an \(\mathcal{A}\)-C3 module where \(\mathcal{A}\) is the class of all simple submodules of \(M\). Moreover, let \(R\) be a right artinian ring and \(\mathcal{A}\), a class of right \(R\)-modules with local 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.