Module-theoretic generalization of commutative von Neumann regular rings

Anderson, Derek T.; Chun, Sangmin; Juett, J. R.

doi:10.1080/00927872.2019.1593427

Cited by 10 publications

(9 citation statements)

References 25 publications

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“…(1)⇔(2) Reduced rings are reversible and hence have IFP. By Lemma 1, (2) follows from (1). Conversely, let a n = 0.…”

Section: Introductionmentioning

confidence: 94%

Characterizations of regular modules

Kimuli,

Ssevviiri

2023

Preprint

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Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.

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“…(1)⇔(2) Reduced rings are reversible and hence have IFP. By Lemma 1, (2) follows from (1). Conversely, let a n = 0.…”

Section: Introductionmentioning

confidence: 94%

Characterizations of regular modules

Kimuli,

Ssevviiri

2023

Preprint

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“…The concept of von Neumann regular rings and its generalizations have drawn considerable interest and have been widely studied by many authors. See, for example, [3], [11] and [12]. Now, in the following, we characterize all rings over which every proper ideal is almost 1-absorbing prime ideal.…”

Section: Characterization Of φ-1-absorbing Prime Idealsmentioning

confidence: 99%

“…So suppose that a ∈ m.Then by Theorem 6, every proper ideal of R m is almost 1-absorbing prime. Since a 3 1 ∈ ( a 3 1 ) and ( a 3 1 ) is almost 1-absorbing prime, we have either a 2 1 ∈ ( a 3 1 ) or a 3 1 ∈ ( a 3 1 ) 2 , which implies that (a 3 ) m = (a 4 ) m . Since (a 3 ) m = (a 4 ) m for each maximal ideal m of R, we have (a 3 ) = (a 4 ) and thus (a 3 ) = (a 3 ) 2 .…”

Section: Characterization Of φ-1-absorbing Prime Idealsmentioning

confidence: 99%

On $ϕ$-1-Absorbing Prime Ideals

Yıldız

Teki̇r

Koç

2020

Preprint

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In this paper, we introduce φ-1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity 1 = 0 and φ : I(R) → I(R) ∪ {∅} be a function where I(R) is the set of all ideals of R. A proper ideal I of R is called a φ-1-absorbing prime ideal if for each nonunits x, y, z ∈ R with xyz ∈ I − φ(I), then either xy ∈ I or z ∈ I. In addition to give many properties and characterizations of φ-1-absorbing prime ideals, we also determine rings in which every proper ideal is φ-1-absorbing prime.

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“…So far, there have been many generalizations of this concept. See, for example, [19], [2] and [6]. Now, we characterize von Neumann regular rings in terms of φ-2-absorbing quasi primary ideals.…”

Section: Characterization Of φ-2-absorbing Quasi Primary Idealsmentioning

confidence: 99%

Generalization of 2-absorbing quasi primary ideals

Uğurlu¹,

Teki̇r²,

Koç³

2020

Preprint

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In this article, we introduce and study the concept of φ-2-absorbing quasi primary ideals in commutative rings. Let R be a commutative ring with a nonzero identity and L(R) be the lattice of all ideals of R. Suppose that φ :In addition to giving many properties of φ-2-absorbing quasi primary ideals, we also use them to characterize von Neumann regular rings.Keywords. weakly 2-absorbing quasi primary ideal, φ-2-absorbing quasi primary ideal, von Neumann regular ring.

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Module-theoretic generalization of commutative von Neumann regular rings

Cited by 10 publications

References 25 publications

Characterizations of regular modules

Characterizations of regular modules

On $ϕ$-1-Absorbing Prime Ideals

Generalization of 2-absorbing quasi primary ideals

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