Multiplication modules over non-commutative rings

Tuganbaev, A. A.

doi:10.1070/sm2003v194n12abeh000788

Cited by 14 publications

(7 citation statements)

References 16 publications

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“…An R-semimodule M is said to be multiplication semimodule if for every subsemimodule N of M there exists an ideal I of R such that N = IM . Then N = (N : M )M [7,25,31].…”

Section: Preliminariesmentioning

confidence: 99%

Properties of the subtractive prime spectrum of a semimodule

Han

HAN>

PAE>

2023

Hacettepe Journal of Mathematics and Statistics

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For a top semimodule over a semiring with zero and nonzero identity, this paper studies the interplay between topological properties of the subtractive prime spectrum and algebraic properties of the semimodule. We prove that the subtractive prime spectrum of the subtractively finitely generated top semimodule is a compact space, and establish necessary and sufficient conditions for the top semimodule to be subtractively finitely generated. For a multiplication semimodule over a commutative semiring, we prove that the radical of a subtractive subsemimodule coincides with its subtractive radical, that every proper subtractive subsemimodule is contained in a subtractive prime subsemimodule, that the multiplication semimodule is subtractively finitely generated iff its subtractive prime spectrum is a compact space, that in the subtractive prime spectrum, the intersection of finitely many basic open sets is compact, and that the subtractive prime spectrum of the subtractively finitely generated multiplication semimodule is a spectral space.

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“…An R-semimodule M is said to be multiplication semimodule if for every subsemimodule N of M there exists an ideal I of R such that N = IM . Then N = (N : M )M [7,25,31].…”

Section: Preliminariesmentioning

confidence: 99%

Properties of the subtractive prime spectrum of a semimodule

Han

HAN>

PAE>

2023

Hacettepe Journal of Mathematics and Statistics

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“…Let R be a ring. An R-module M is called a multiplication module if every submodule N of M has the form IM , for some ideal I of R, see [10]. We know that M is a multiplication R-module if and only if N = (N : R M )M , for every submodule N of M .…”

Section: Classical Weakly Prime Submodules Of a Multiplication Modulementioning

confidence: 99%

On Classical Weakly Prime Submodules

Jamali

Jahani-Nezhad

2022

FU Math Inform

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The aim of this paper is to introduce the concept of classical weakly prime submodules which is the generalization of the notion of weakly classical prime submodules to modules over arbitrary noncommutative rings. We study some properties of classical weakly prime submodules and investigate their structure in different classes of modules. Also, the structure of such submodules of modules over duo rings is completely described. We investigate some properties of classical weakly prime submodules of multiplication modules.

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“…If R is a right top ring, then it is a quasi-duo ring (see Zhang et al, 2006, Corollary 2.8). Top modules and top rings have been investigated in McCasland et al (1997), Tuganbaev (2003, Zhang (1999), Zhang and Tong (2000), and Zhang et al (2004Zhang et al ( , 2006.…”

Section: Introductionmentioning

confidence: 99%

Spectrum of a Noncommutative Ring

Zhang

Tong

Wang

2006

Communications in Algebra

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R is any ring with identity. Let Spec r R resp. Spec R be the set of all prime right ideals resp. all prime ideals of R and let U r eR = P ∈ Spec r R e P . In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r R with weak Zariski topology . A ring R is called Abelian if all idempotents in R are central see Goodearl, 1991 . A ring R is called 2-primal if every nilpotent element is in the prime radical of R see Lam, 2001 . It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen i.e., closed and open sets in Spec r R . And the following results are obtained for any ring R: 1 For any clopen set U in Spec r R , there is an idempotent e in R such that U = U r eR . 2 If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r eR is a clopen set in Spec r R . 3 Spec r R is connected if and only if Spec R is connected.

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Multiplication modules over non-commutative rings

Cited by 14 publications

References 16 publications

Properties of the subtractive prime spectrum of a semimodule

Properties of the subtractive prime spectrum of a semimodule

On Classical Weakly Prime Submodules

Spectrum of a Noncommutative Ring

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