Nonnil-coherent rings

bacem, Khoualdia; Benhissi, Ali

doi:10.1007/s13366-015-0260-8

Cited by 12 publications

(4 citation statements)

References 6 publications

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“…Recall from [7] that a φ-ring R is called nonnil-Noetherian if any nonnil ideal of R is finitely generated. Recall from [4] that a φ-ring R is called nonnil-coherent if any finitely generated nonnil ideal of R is finitely presented. A φ-ring R is nonnil-coherent if and only if any direct product of φ-flat modules is φ-flat, if and only if R I is φ-flat for any indexing set I (see [4,Theorem 2.4]).…”

Section: Nonnil-injective Modules and Nonnil-fp-injective Modulesmentioning

confidence: 99%

“…Recall from [4] that a φ-ring R is called nonnil-coherent if any finitely generated nonnil ideal of R is finitely presented. A φ-ring R is nonnil-coherent if and only if any direct product of φ-flat modules is φ-flat, if and only if R I is φ-flat for any indexing set I (see [4,Theorem 2.4]). Now we give a new characterization of nonnil-coherent rings utilizing the preenveloping properties of φ-flat modules.…”

Section: Nonnil-injective Modules and Nonnil-fp-injective Modulesmentioning

confidence: 99%

“…Let {F i |i ∈ I} be a family of φ-flat modules. Then i∈I F i is φ-flat by [4,Theorem 2.4]. By (7), the class of φ-flat modules is closed under submodules.…”

Section: Recall That a Regular Idealmentioning

confidence: 99%

“…They also showed that a φ-ring R is φ-Dedekind if and only if R is nonnil-Noetherian and R m is a discrete φ-chained ring for any maximal ideal m of R, if and only if R is nonnil-Noetherian, φ-integral closed and of Krull dimension ≤ 1, if and only if R/Nil(R) is a Dedekind domain. Some generalizations of Noetherian domains, coherent domains, Bezout domains and Krull domains to the context of rings that are in the class H are also introduced and studied (see [1,2,4,7,8]).…”

mentioning

confidence: 99%

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Some Remarks on $ϕ$-Dedekind rings and $ϕ$-Prufer rings

Zhang¹,

Qi²

2021

Preprint

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In this paper, the notions of nonnil-injective modules and nonnil-FPinjective modules are introduced and studied. Especially, we show that a φ-ring R is an integral domain if and only if any nonnil-injective (resp., nonnil-FP-injective) module R-module is injective (resp., FP-injective). Some new characterizations of φ-von Neumann regular rings, nonnil-Notherian rings and nonnil-coherent rings are given. We finally characterize φ-Dedekind rings and φ-Prüfer rings in terms of φ-flat modules, nonnil-injective modules and nonnil-FP-injective modules.

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Section: Nonnil-injective Modules and Nonnil-fp-injective Modulesmentioning

confidence: 99%

Section: Nonnil-injective Modules and Nonnil-fp-injective Modulesmentioning

confidence: 99%

“…Let {F i |i ∈ I} be a family of φ-flat modules. Then i∈I F i is φ-flat by [4,Theorem 2.4]. By (7), the class of φ-flat modules is closed under submodules.…”

Section: Recall That a Regular Idealmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Some Remarks on $ϕ$-Dedekind rings and $ϕ$-Prufer rings

Zhang¹,

Qi²

2021

Preprint

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Nonnil-FP-injective and nonnil-FP-projective dimensions and nonnil-semihereditary rings

Haddaoui,

Kim,

Mahdou

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper, we introduce the concept of nonnil-FP-injective dimension for both modules and rings. We explore the characterization of strongly $$\phi $$ ϕ -rings that have a nonnil-FP-injective dimension of at most one. We demonstrate that, for a nonnil-coherent, strongly $$\phi $$ ϕ -ring R, the nonnil-FP-injective dimension of R corresponds to the supremum of the $$\phi $$ ϕ -projective dimensions of specific families of R-modules. We also define self-nonnil-injective rings as $$\phi $$ ϕ -rings that act as nonnil semi-injective modules over themselves and establish the equivalence between a strongly $$\phi $$ ϕ -ring R being $$\phi $$ ϕ -von Neumann regular and R being both nonnil-coherent and self-nonnil semi-injective. Furthermore, we extend the notion of semihereditary rings to $$\phi $$ ϕ -rings, coining the term ‘nonnil-semihereditary’ to describe rings where every finitely generated nonnil ideal is u-$$\phi $$ ϕ -projective. We provide several characterizations of nonnil-semihereditary rings through various conceptual lenses. Our study also includes an investigation of the transfer of the nonnil-semihereditary property in trivial ring extensions. Additionally, we define the nonnil-FP-projective dimension for modules and rings, showing that for any strongly $$\phi $$ ϕ -ring, a nonnil-FP-projective dimension of zero is indicative of the ring being nonnil-Noetherian. We also ascertain that, for a strongly $$\phi $$ ϕ -ring R, its nonnil-FP-projective dimension is the supremum of the NFP-projective dimensions across different families of R-modules. Lastly, we provide numerous examples to illustrate our results.

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