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In this paper, we introduce and study the notions of $ M $-strongly hollow and $ M $-PS-hollow ideals where $ M $ is a module over a commutative ring $ R $. These notions are generalizations of strongly hollow ideals. We investigate some properties and characterizations of $ M $-strongly hollow ($ M $-PS-hollow) ideals. Then we define and study a topology on the set of all $ M $-PS-hollow ideals of a commutative ring $ R $. We investigate when this topological space is irreducible, Noetherian, $ T_{0} $, $ T_{1} $ and spectral space.

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 $$\hbox {Spec}^{s}(M)$$ 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 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.

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.

Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P i (1 ≤ i ≤ k) such that if L is a proper submodule of M...

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.

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 introduce the dual Zariski topology on the set of second submodules of M, denoted by Specs(M), for an R‐module M. We give some relationships between Specs(M) and Spec(R/Ann(M)). By using this topological space, we give some characterizations of rings and modules.

In this paper, we study on the module M = R⊕R over a commutative ring R with identity. After characterizing the radical of a submodule N of M, we give some conditions for N to satisfy the radical formula. In particular, we show that R⊕R satisfy the radical formula if R is an arithmetical ring.