Document Type : Review Article
Authors
1 Department of Science and Humanities (Mathematics) National Institute of Technology Nagaland, Chumukedima-797103, India
2 National Institute of Technology Nagaland
3 Department of Science and Humanities (Mathematics) National Institute of Technology Nagaland, Chumukedima-797103
Abstract
This survey presents a comprehensive and systematic overview of the theory of chain conditions on modules and rings with particular emphasis on injective and quasi-injective modules, as well as associated structural properties of rings. We cover classical and modern perspectives on ascending and descending chain conditions (acc and dcc) on submodules, essential submodules, and quotient modules. Key classes such as Noetherian, Artinian, seminoetherian, and isonoetherian modules and rings are explored in detail. We review the theory of quasi-injective modules under chain conditions, including characterization
theorems, endomorphism ring decompositions, and their connections to quasi-Frobenius rings. The notions of essential Noetherian and essential Artinian modules and their significance in generalizing classical results. Finally, we touch upon divisibility properties in submodule chains, power series ring extensions, projective modules over semilocal rings, and related topics, highlighting open problems and future research directions.
Keywords
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