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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.

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.

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.

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.

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.

We compute the automorphism groups of some quantized algebras, including tensor products of quantum Weyl algebras and some skew polynomial rings., Comment: 38 pages

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 m*-system and describe the second radical of submodules in terms of m*-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 R/P 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. (C) 2013 Elsevier Inc. All rights reserved.

In this paper, we consider the free R-module R circle plus R, where R is an arbitrary commutative ring with identity. We give a full characterization for prime submodules of R circle plus R and a useful primeness test for a finitely generated submodule of R circle plus R. We study the existence of primary decomposition of a submodule of R circle plus R and characterize the minimal primary decomposition. As applications of our results, we give some examples of primary decompositions in R circle plus R.