On the Notion of Strong Irreducibility and Its Dual

Abuhlail, Jawad; Lomp, Christian

doi:10.1142/s0219498813500126

Cited by 5 publications

(6 citation statements)

References 28 publications

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“…The notion of a strongly hollow submodule was introduced by Abuhlail in [6], as dual to that of strongly irreducible submodules. The notion was generalized to general lattices and investigated by Abuhlail and Lomp in [3].…”

Section: ([12]mentioning

confidence: 99%

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PS-Hollow Representations of Modules over Commutative Rings

Abuhlail¹,

Hroub²

2018

Preprint

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Let R be a commutative ring and M a non-zero R-module. We introduce the class of pseudo strongly hollow submodules (PS-hollow submodules, for short) of M. 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. * MSC2010: Primary 13C13.

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Section: ([12]mentioning

confidence: 99%

“…We generalized the notion of a strongly hollow element of a lattice investigated by Abuhlail and Lomp in [3] to a strongly hollow element of a lattice with an action from a poset. Moreover, we introduced its dual notion of a pseudo strongly irreducible element which is a generalization of the notion of a strongly irreducible element.…”

Section: 3mentioning

confidence: 99%

PS-Hollow Representations of Modules over Commutative Rings

Abuhlail¹,

Hroub²

2018

Preprint

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“…1. An element x ∈ L\{1} is said to be: irreducible [7] iff for any a, b ∈ L with a ∧ b = x, we have a = x or b = x;…”

Section: Introductionmentioning

confidence: 99%

“…hollow iff whenever for any a, b ∈ L with x = a ∨ b, we have x = a or x = b; strongly hollow [7] iff for any a, b ∈ L with x ≤ a ∨ b, we have x ≤ a or x ≤ b.…”

Section: Introductionmentioning

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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Strongly hollow elements of commutative rings

Rostami

2020

J. Algebra Appl.

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In this paper, the notion of completely strongly hollow ideals (respectively, strongly hollow elements) of a commutative ring as a generalization of strongly hollow ideals (respectively, local idempotents) is introduced and some related properties are investigated. Also, some characteristics of completely strongly hollow ideals and strongly hollow elements of some special classes of commutative rings are given. Then, we consider the decomposition of nonzero ideals of a ring into a product (respectively, (finite) sum) of completely strongly hollow ideals. Some other results are obtained too.

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On the Notion of Strong Irreducibility and Its Dual

Cited by 5 publications

References 28 publications

PS-Hollow Representations of Modules over Commutative Rings

PS-Hollow Representations of Modules over Commutative Rings

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

Strongly hollow elements of commutative rings

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