In this paper, we will introduce two generalizations of second submodules of a module over a commutative ring and explore some basic properties of these classes of modules
Let R be a commutative ring and M an R-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of M) and investigate the conditions under which the number of maximal second submodules of M is finite. Furthermore, we introduce the concept of coisolated submodules of M and obtain some related characterizations.
Let R be a commutative ring and M be an R-module. The second spectrum Spec s(M) of M is the collection of all second submodules of M. We topologize Spec s(M) with Zariski topology, which is analogous to that for Spec (M), and investigate this topological space. For various types of modules M, we obtain conditions under which Spec s(M) is a spectral space. We also investigate Spec s(M) with quasi-Zariski topology.
Let R be a ring with an identity (not necessary commutative) and let M be a left R-module. In this paper we will introduce the concept of a comultiplication R-module and we will obtain some related results.
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