scholarly journals SOME REMARKS ON THE CLASSICAL PRIME SPECTRUM OF MODULES

2021 ◽  
pp. 015
Author(s):  
Alireza Abbasi ◽  
Mohammad Hasan Naderi
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Prime Submodules ◽  
Prime Submodule

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P, for a,b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum of M, Cl.Spec(M), is defined to be the set of all classical prime submodules of M. We say M is classical primefule if M = 0, or the map ψ from Cl.Spec(M) to Spec(R/Ann(M)), defined by ψ(P) = (P : M)/Ann(M) for all P ∈ Cl.Spec(M), is surjective. In this paper, we study classical primeful modules as a generalisation of primeful modules. Also we investigate some properties of a topology that is defined on Cl.Spec(M), named the Zariski topology.

On the prime spectrum of an le-module

2020 ◽  
pp. 2150220
Author(s):  
M. Kumbhakar ◽  
A. K. Bhuniya
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Closed Subset ◽  
Quotient Ring ◽  
Acta Math ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Generic Point ◽  
Prime Submodule

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if [Formula: see text] contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring [Formula: see text] induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

Zariski Topologies on Graded Ideals

2021 ◽  
Vol 78 (1) ◽  
pp. 215-224
Author(s):  
Malik Bataineh ◽  
Azzh Saad Alshehry ◽  
Rashid Abu-Dawwas
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Algebraic Properties

Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

2021 ◽  
Vol 10 (11) ◽  
pp. 3479-3489
Author(s):  
K. Al-Zoubi ◽  
M. Al-Azaizeh
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Closed Subset ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

scholarly journals On graded Ie-prime submodules of graded modules over graded commutative rings

2021 ◽  
Vol 110 (124) ◽  
pp. 47-55
Author(s):  
Shatha Alghueiri ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I =?g?G Ig a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded Ie-prime submodule of M if whenever rg ? h(R) and mh ? h(M) with rgmh ? N ? IeN, then either rg ? (N :R M) or mh ? N.

scholarly journals φ-PRIME SUBMODULES

2009 ◽  
Vol 52 (2) ◽  
pp. 253-259 ◽  
Author(s):  
NASER ZAMANI
Keyword(s):  
Commutative Ring ◽  
Prime Submodules ◽  
Prime Submodule

AbstractLet R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

scholarly journals Approximaitly Prime Submodules and Some Related Concepts

2019 ◽  
Vol 32 (2) ◽  
pp. 103
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammad Ali
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Research Note ◽  
Prime Radical ◽  
Basic Properties ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Definition Of

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule  of an -module  is called an approximaitly prime submodule of  (for short app-prime submodule), if when ever , where , , implies that either  or . So, an ideal  of a ring  is called app-prime ideal of  if   is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.

The Zariski Topology on the Prime Spectrum of a Module over Noncommutative Rings

2009 ◽  
Vol 16 (04) ◽  
pp. 691-698 ◽  
Author(s):  
Ünsal Tekir
Keyword(s):  
Associative Ring ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Noncommutative Rings ◽  
Prime Submodules

Let R be an associative ring with identity and M an R-module. Let Spec (M) be the set of all prime submodules of M. We topologize Spec (M) with the Zariski topology and prove some useful results.

Attaching a topological space to a module

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3171-3180
Author(s):  
Dawood Hassanzadeh-Lelekaami
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Zariski Topology ◽  
Prime Spectrum

Let R be a commutative ring with identity and let M be an R-module. We investigate when the strongly prime spectrum of M has a Zariski topology analogous to that for R. We provide some examples of such modules.

scholarly journals Approximaitly Quasi-Prime Submodules and Some Related Concepts

2019 ◽  
Vol 11 (2) ◽  
pp. 54-62
Author(s):  
Haibat K. Mohammadali ◽  
Ali Sh. Ajeel
Keyword(s):  
Commutative Ring ◽  
Basic Properties ◽  
Prime Submodules ◽  
Unitary Module ◽  
Prime Submodule ◽  
Properties Characterization

“Let  be a commutative ring with identity and  is a left unitary -module. A proper submodule  of  is called a quasi-prime submodule, if whenever , where ,  implies that either  or ”. As a generalization of a quasi-prime submodules, in this paper we introduce the concept of approximaitly quasi-prime submodules, where a proper submodule  of  is an approximaitly quasi-prime submodule, if whenever , where ,  implies that either  or , where  is the intersection of all essential submodules of . Many basic properties, characterization and examples of this concept are given. Furthermore, we study the behavior of approximaitly quasi-prime submodules under -homomorphisms. Finally, we introduced characterizations of approximaitly quasi-prime submodule in class of multiplication modules.

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