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Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and let M be an R-module. In this paper, we introduce and investigate some properties of the notion of S-2-absorbing second submodules of M as a generalization of S-second submodules and strongly 2-absorbing second submodules of M. Also, we obtain some results concerning S-2-absorbing submodules of M. [ABSTRACT FROM AUTHOR]

The concept of comultiplication (as a dual notion of multiplication) modules was introduced and studied by H. Ansari-Toroghy and F. Farshadifar in 2007. This notion has obtained a great attention by many authors and now there is a considerable amount of research concerning this class of modules. The main purpose of this paper is to collect these results and provide a useful source for those who are interested in research in this field.

Let R be a commutative ring with identity and M be an R-module. Let ψ : S (M) → S (M) ∪ { ∅ } be a function, where S(M) denote the set of all submodules of M. The main purpose of this paper is to introduce and study the notion of ψ -second submodules of an R-module M. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, I an ideal of R, and M be an R-module. In this paper, we will introduce the concept of I-second submodules of M as a generalization of second submodules of M and obtain some related results. [ABSTRACT FROM AUTHOR]

The concept of comultiplication (as a dual notion of multiplication) modules was introduced and studied by H. Ansari-Toroghy and F. Farshadifar in 2007. This notion has obtained a great attention by many authors and now there is a considerable amount of research concerning this class of modules. The main purpose of this paper is to collect these results and provide a useful source for those who are interested in research in this field. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and Specs(M) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of Specs(M) with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for Specs(M) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by , on equipped with the dual Zariski topology of M, where N is an R-module. We give a characterization of the sections of the sheaf in terms of the ideal transform module. We present some interrelations between algebraic properties of N and the sections of . We obtain some morphisms of sheaves induced by ring and module homomorphisms. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we present a number of new results for the second classical Zariski topology on Specs(M) for a right R-module M. We obtain a characterization of semisimple modules by using the second spectrum of a module. We prove that if R is a ring such that every right primitive factor of R is right artinian, then every non-zero submodule of a second right R-module M is second if and only if M is a fully prime module. We give some equivalent conditions for Specs(M) to be a Hausdorff space or T1-space when the right R-module M has certain algebraic properties. We obtain characterizations of commutative Quasi-Frobenius and artinian rings by using topological properties of the second classical Zariski topology. We give a full characterization of the irreducible components of Specs(M) for a non-zero injective right module M over a ring R such that every prime factor of R is right or left Goldie. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M be an R-module. Let Specs(M) be the set of all second submodules of M. In this article, we topologize Specs(M) with Zariski and classical Zariski topologies and study the classes of all modules whose second spectrum have the surjective or injective natural map. Moreover, we investigate the interplay between the algebraic properties of M and the topological properties of Specs(M). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M an R-module. Let $$Spec^s(M)$$ be the collection of all second submodules of M. In this article, we consider a new topology on $$Spec^s(M)$$ , called the second classical Zariski topology, and investigate the interplay between the module theoretic properties of M and the topological properties of $$Spec^s(M)$$ . Moreover, we study $$Spec^s(M)$$ from point of view of spectral space. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected. [ABSTRACT FROM AUTHOR]

Let M be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Specs(M) of second submodules of M and the spectrum Specc(M) of coprime submodules of M, study several properties of these spaces and investigate their interplay with the algebraic properties of M. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M be an R-module. The second spectrum s(M) of M is the collection of all second submodules of M. We topologize s(M) with Zariski topology, which is analogous to that for (M), and investigate this topological space. For various types of modules M, we obtain conditions under which s(M) is a spectral space. We also investigate s(M) with quasi-Zariski topology. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we study the second radical of a module over an arbitrary ring R as the dual notion of the prime radical of a module. We give some properties of the second radical and determine the second radical of some modules. We define the notion of -system and describe the second radical of submodules in terms of -systems. We investigate when the second radical of a module M is equal to the socle of M. In particular, we give a characterization of the socle of a noetherian module over a ring R such that the ring is right artinian for every right primitive ideal P by using the concept of second radical. We also give a characterization of right quasi-duo artinian rings by using the second radical of an injective module. [Copyright &y& Elsevier]