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Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.

The purpose of this paper is to introduce the concept of 2-absorbing coprimary submodule of a module as a generalization of strongly 2-absorbing second submodules and investigate some properties of this submodule class. We study the structure of 2-absorbing coprimary submodules of comultiplication modules, fully coidempotent modules and modules over a Noetherian ring. We also investigate some interrelations with 2-absorbing coprimary submodules, strongly 2-absorbing second submodules and strongly 2-absorbing secondary submodules.

Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Specs(M). For each prime ideal p of R we define Specsp(M) := {S? Specs(M) : annR(S) = p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Specs p(M)? 0 and Q = ? S2Specsp(M)S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Specs(M) with the dual Zarsiki topology. Also, we topologize u.Specs(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Specs(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster?s characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Specs(M) with the dual Zariski topology is a spectral space.

Let R be a commutative ring with identity and τ be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of τ-comultiplication module. We present several propert...

Let R be a commutative ring with identity and $$\hbox {Spec}^{s}(M)$$ denote the set all second submodules of an R-module M. In this paper, we investigate various properties of $$\hbox {Spec}^{s}(M)$$ 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 $$\hbox {Spec}^{s}(M)$$ 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.

In this paper, we investigate interior operations on submodules and introduce a new interior operation by using a certain submodule class. Let R be a commutative ring with identity and I be a set of ideals of R. We define I-second submodules and I-interior of a submodule. We show that second, secondary and strongly second submodules are special types of I-second submodules. We investigate several properties of I-interiors of submodules and give a concrete expression of I-interior of a submodule of an Artinian module. We use the concept of I-interior of a submodule to find some results on I-second submodules and attached primes of an Artinian module.Mathematics Subject Classification (2010): 13A15, 13C13, 13C99.Keywords: I-second submodule, interior operation, I-interior of a submodule, attached prime ideal

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.

In this paper, we extend the definition of weakly second submodule of a module over a commutative ring to a module over an arbitrary ring. First, we investigate some properties of weakly second submodules. We define the notion of weakly second radical of a submodule and determine the weakly second radical of some modules. We also define the notion of weak m*-system and characterize the weakly second radical of a submodule in terms of weak m*-systems. Then we introduce and study a topology on the set of all weakly second submodules of a module. We give some results concerning irreducible subsets, irreducible components and compactness of this topological space. Finally, we investigate this topological space from the point of view of spectral spaces.

In this paper, we study on the concept of strongly 2-absorbing second submodule which is a dual notion of 2-absorbing submodule and a generalization of second submodule. We give some properties and characterizations of this submodule class and investigate the relationships with second and secondary submodules.

Let R be a commutative ring with identity and Spec(5)(M) denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by O(N, M), on Spec(5)(M) equipped with the dual Zariski topology of M, where N is an R-module. We give a characterization of the sections of the sheaf O(N, M) in terms of the ideal transform module. We present some interrelations between algebraic properties of N and the sections of O(N, M). We obtain some morphisms of sheaves induced by ring and module homomorphisms.

We use the discriminant to determine the automorphism groups of some noncommutative algebras, and we prove that a family of noncommutative algebras has tractable automorphism groups. There is a long history and an extensive study of the automorphism groups of algebras. Determining the full automorphism group of an algebra is generally a no- toriously difficult problem. For example, the automorphism group of t polynomial ring of three variables is not yet understood, and a remarkable result in this direction is given by Shestakov-Umirbaev (SU) which shows the Nagata automorphism is a wild automorphism. Since 1990s, many researchers have been successfully comput- ing the automorphism groups of interesting infinite-dimensional noncommutative algebras, including certain quantum groups, generalized quantum Weyl algebras, skew polynomial rings and many more - see (AlC, AlD, AnD, BJ, GTK, SAV), which is only a partial list. Recently, by using a rigidity theorem for quantum tori, Yakimov has proved the Andruskiewitsch-Dumas conjecture and the Launois- Lenagan conjecture in (Y1, Y2), each of which determines the automorphism group of a family of quantized algebras with parameter q being not a root of unity. A uniform approach to both the Andruskiewitsch-Dumas conjecture and the Launois- Lenagan conjecture is provided in a preprint by Goodearl-Yakimov (GY). These beautiful results, as well as others, motivated us to look into the automorphism groups of noncommutative algebras.

Let R be a commutative ring with identity and S pecs(M) (resp. Min(M)) denote the set of all second (resp. minimal) submodules of a non-zero R-module M. In this paper, we investigate several properties of the subspace topology on Min(M) induced by the dual Zariski on S pecs(M) and determine some cases in which Min(M) is a max-spectral space.

In this paper, we introduce and study graded 2-absorbing and graded weakly 2-absorbing ideals of a graded ring which are different from 2-absorbing and weakly 2-absorbing ideals. We give some properties and characterizations of these ideals and their homogeneous components. We investigate graded (weakly) 2-absorbing ideals of $R_{1}\times R_{2}$ where $R_{1}$ and $R_{2}$ are two graded rings.

In this paper we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are different from second and coprimary modules over arbitrary-graded rings. We list some properties and characterizations of gr-second and gr-coprimary modules and also study graded prime submodules of modules with gr-coprimary decompositions. We also deal with graded secondary representations for graded injective modules over commutative-graded rings. By using the concept of \(\sigma \)-suspension \((\sigma )M\) of a graded module \(M,\) we prove that a graded injective module over a commutative graded Noetherian ring has a graded secondary representation.

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.