On topological lattices and their applications to module theory

Abuhlail, Jawad; Lomp, Christian

doi:10.1142/s0219498816500468

Cited by 3 publications

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References 24 publications

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“…It is worth mentioning that Theorem 2.6 recovers several results of Abuhlail on such 1-1 correspondences for L = LAT ( R M) (e.g. [2], [3], [4]) and Abuhlail/Lomp [1] as special cases (some of these results are recovered under conditions weaker than those assumed in the original results for the different spectra of modules).…”

Section: Introductionsupporting

confidence: 65%

“…If R M has the property that H(A) is first whenever A is irreducible, then SH(H (L)) ⊆ X and so the 1-1 correspondences of Theorem 2.6 hold. This was proved under the same condition in [1].…”

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confidence: 65%

“…Then there are x ′ , y ′ ∈ C (L) such that x ∨x ′ = 1, x ∧ x ′ = √ 0, y ∨y ′ = 1 and y ∧ y ′ = √ 0. One can check that x ∧ y and x ∨y are also in C ′ with the corresponding elements x ′ ∨y ′ and x ′ ∧ y ′ respectively (recall that if L is X -top then C (L ) is distributive by [1,Theorem 2.2]).…”

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confidence: 99%

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Zariski-like topologies for lattices with applications to modules over associative rings

Abuhlail

Hroub

2019

J. Algebra Appl.

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We study Zariski-like topologies on a proper class X L of a complete lattice L = (L, ∧, ∨, 0, 1). We consider X with the so called classical Zariski topology (X , τ cl ) and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that L is X -top iffis a topology. We study the interplay between the algebraic properties of an X -top complete lattice L and the topological properties of (X , τ cl ) = (X , τ). Our results are applied to several spectra which are proper classes of L := LAT ( R M) where M is a left module over an arbitrary associative ring R (e.g. the spectra of prime, coprime, fully prime submodules) of M as well as to several spectra of the dual complete lattice L 0 (e.g. the spectra of first, second and fully coprime submodules of M). * MSC2010: Primary 06A15; Secondary 16D10, 13C05, 13C13, 54B99.The spectrum Spec(R) of prime ideals of a commutative ring R attains the so called Zariski topology in which the closed sets are the varietiesThis topology is compact, T 0 but almost never T 2 , and the closed points correspond to the maximal ideals. The Zariski topology proved to be very important in two main aspects: in Algebraic Geometry and in Commutative Algebra. In particular, it provided an efficient tool for studying the algebraic properties of a commutative ring R by investigating the corresponding topological properties of Spec(R) [9].Motivated by this, there were many attempts to define Zariski-like topologies on the spectra of prime-like submodules of a given left module M over a (not necessarily commutative) ring R. This resulted at the first place in several different notions of prime submodules of R M which reduced to the notion of a prime ideal for the special case M = R, a commutative ring (e.g. [21]). The work in this direction was almost limited to studying these prime-like submodules and their duals (the coprime-like submodules) as well as to the families of prime ideals corresponding to them from a purely algebraic point of view. One of the obstacles was that not every module M over a (commutative) ring R has the property that Spec(M) attains a Zariski-like topology: the proposed closed varieties {V (N) | N ∈ LAT ( R M)} are not necessarily closed under finite unions. Modules for which this last condition is satisfied were investigated, among others, by R. L. McCasland and P. F. Smith (e.g.[17], [16]) and called top modules. However, even such modules were studied from a purely algebraic point of view and the associated Zariski-like topologies were not well studied till about a decade ago. In [6], Abuhlail introduced a Zariskilike topology on the spectrum of fully coprime subcomodules of a given comodule M of a coring C over an associative ring R and studied the interplay between the algebraic properties of M and the topological properties of that Zariski-like topology (see also [5]).Later, in a series of papers ([3], [4], [2]), Abuhlail introduced and investigated several Zariskili...

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Section: Introductionsupporting

confidence: 65%

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confidence: 65%

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Zariski-like topologies for lattices with applications to modules over associative rings

Abuhlail

Hroub

2019

J. Algebra Appl.

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PS-hollow representations of modules over commutative rings

Abuhlail

Hroub

2021

J. Algebra Appl.

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Let [Formula: see text] be a commutative ring and [Formula: see text] a nonzero [Formula: see text]-module. We introduce the class of pseudo-strongly[Formula: see text]PS[Formula: see text]-hollow submodules of [Formula: see text]. Inspired by the theory of modules with secondary representations, we investigate modules which can be written as finite sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of minimal PS-hollow strongly representations of modules over Artinian rings.

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Generalizations of strongly hollow ideals and a corresponding topology

Çeken¹,

Yüksel

2021

MATH

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<abstract><p>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.</p></abstract>

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On topological lattices and their applications to module theory

Cited by 3 publications

References 24 publications

Zariski-like topologies for lattices with applications to modules over associative rings

Zariski-like topologies for lattices with applications to modules over associative rings

PS-hollow representations of modules over commutative rings

Generalizations of strongly hollow ideals and a corresponding topology

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