1,720,969 research outputs found
On a subspace of dual Zariski topology
No full textLet R be a commutative ring with identity and S pee (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 pee(M) and determine some cases in which Min(M) is a max-spectral space
MODULES AND THE SECOND CLASSICAL ZARISKI TOPOLOGY
No full textLet R be an associative ring with identity and Spec(s)(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 Spec(s)(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 Spec(s)(M) to be a Hausdorff space or T-i-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 Spec(s)(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.Scientific Research Project Administration of Akdeniz UniversityThe second author was supported by the Scientific Research Project Administration of Akdeniz University
On the interior of a submodule with respect to a set of ideals
No full textIn 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 be a set of ideals of R. We define -second submodules and -interior of a submodule. We show that second, secondary and strongly second submodules are special types of -second submodules. We investigate several properties of -interiors of submodules and give a concrete expression of - interior of a submodule of an Artinian module. We use the concept of -interior of a submodule to find some results on -second submodules and attached primes of an Artinian module
On S-semisecond and S-semiprime submodules
No full textLet R be a commutative ring with identity, S be a multiplicatively closed subset of R. In this article we introduce and investigate the concept of S-semisecond submodule as a generalization of semisecond and S-second submodules. We also give some results on S-semiprime submodules and investigate some interrelations between S-semiprime and S-semisecond submodules. In addition, to give many examples and characterizations of S-semisecond submodules, we characterize a certain class of semisecond submodules in terms of S-semisecond submodules. We also determine several characterizations of modules M in which every submodule N of M with annR(N)boolean AND S=& empty; is S-semisecond
On S-second spectrum of a module
No full textLet R be a commutative ring with identity, S be a multiplicatively closed subset of R. A submodule N of an R-module M with ann(R)(N) boolean AND S = empty set is called an S-second submodule of M if there exists a fixed s is an element of S, and whenever rN subset of K, where r is an element of R and K is a submodule of M, then either rsN = 0 or sN subset of K. The set of all S-second submodules of M is called S-second spectrum of M and denoted by S-Specs (M). In this paper, we construct and study two topologies on S-Spec(s) (M). We investigate some connections between algebraic properties of M and topological properties of S-Spec(s) (M) such as seperation axioms, compactness, connectedness and irreducibility
On the Upper Dual Zariski Topology
No full text1st Mediterranean International Conference of Pure and Applied Mathematics and Related Areas (MICOPAM) -- OCT 26-29, 2018 -- Akdeniz Univ, Antalya, TURKEYLet 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 Spec(s)(M). For each prime ideal p of R we define Spec(p)(s)(M) := {S is an element of Spec(s)(M) : ann(R)(S) = p g. A second submodule Q ofMis called an upper second submodule if there exists a prime ideal p of R such that Spec(p)(s)(M)not equal (sic) and Q = Sigma S is an element of Spec(p)(s)(M) S. The set of all upper second submodules ofMis called upper second spectrumofMand denoted by u:Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u:Spec(s)(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-moduleM, u:Spec(s)(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u:Spec(s)(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
Dual Zariski Spaces of Modules
No full textLet R be a commutative ring with identity, M be an R-module, L (M ) denote the set of all submodules of M and G subset of L ( M) \ { 0(M) } . For any submodule N of M, we set GV(d) ( N) = { K is an element of G : K subset of N } and G zeta(d) (M ) = { GV(d) ( N) : N is an element of L (M ) } . Consider chi subset of L ( R) \ { R } , where L (R ) is the set of all ideals of R. We set chi V (I ) = { J is an element of chi : I subset of J } and chi zeta (R ) = { chi V (I ) : I is an element of L (R ) } for any ideal I of R. In this paper, we investigate when, for arbitrary chi and G as above, chi zeta (R ) and G zeta(d) (M ) form a topology and a semimodule, respectively. We investigate the structure of G zeta(d) (M ) in the case that it is a semimodule
Comultiplication modules relative to a hereditary torsion theory
No full textLet R be a commutative ring with identity and tau be a hereditary torsion theory on R-Mod. In this article, we introduce and study the concept of tau-comultiplication module. We present several properties and characterizations of tau-comultiplication modules. We also investigate modules for which every submodule has a unique tau-pseudo-complement and prove that every tau-comultiplication module is a module with unique tau-pseudo-complements.Trakya UniversitesiTrakya Universitesi
Generalizations of strongly hollow ideals and a corresponding topology
No full textIn 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
Classical S-Zariski Topology of a Module
No full textLet R be a commutative ring with identity and let S be a multiplicatively closed subset of R. A submodule P of an R-module M with (P:RM)boolean AND S=& empty;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is said to be an S-prime submodule of M if there exists a fixed s is an element of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and whenever am is an element of P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, then sa is an element of(P:RM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} or sm is an element of P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} for each a is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, m is an element of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. The set of all S-prime submodules of M is denoted by SpecS(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. In this paper, we construct and investigate a topology on SpecS(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} which we will call classical S-Zariski topology for an R-module M. We use specific algebraic properties of M to obtain some topological properties such as separation axioms, compactness, connectedness, and irreducibility. We also investigate classical S-Zariski topology from the point of view spectral spaces by using Hochster's characterization.Scientific and Technological Research Council of Turkiye (TUBITAK)Open access funding provided by the Scientific and Technological Research Council of Turkiye (TUBITAK)
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