A new semistar operation on a commutative ring and its applications

Zhou, De Chuan; Kim, Hwankoo; Wang, Fanggui; Chen, Dan

doi:10.1080/00927872.2020.1753060

Cited by 6 publications

(16 citation statements)

References 16 publications

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“…Recently, the authors in [27] introduced the notion of q-operation on a commutative ring R. We give some reviews here. An R-module M is said to be Q-torison-free if Im = 0 with I ∈ Q and m ∈ M can deduce m = 0.…”

Section: Rings With Sr-w-weak Global Dimensions At Most Onementioning

confidence: 99%

A homological characterization of $Q_0$-Prüfer $v$-multiplication rings

Zhang¹

2021

Preprint

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Let R be a commutative ring. An R-module M is called a semi-regular w-flat module if Tor R 1 (R/I, M) is GV-torsion for any finitely generated semi-regular ideal I. In this article, we showed that the class of semiregular w-flat modules is a covering class. Moreover, we introduce the semi-regular w-flat dimensions of R-modules and the sr-w-weak global dimensions of the commutative ring R. Utilizing these notions, we give some homological characterizations of WQ-rings and Q 0 -PvMRs.

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Section: Rings With Sr-w-weak Global Dimensions At Most Onementioning

confidence: 99%

A homological characterization of $Q_0$-Prüfer $v$-multiplication rings

Zhang¹

2021

Preprint

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“…Recently, Zhou et al [24] introduced the notion of q-operations over commutative rings utilizing finitely generated semi-regular ideals. q-operations are semi-star operations which are weaker than w-operations.…”

Section: Introductionmentioning

confidence: 99%

“…q-operations are semi-star operations which are weaker than w-operations. The authors in [24] also proposed τ q -Noetherian rings and study them via moduletic-theoretic point of view, such as τ q -analog of the Hilbert basis theorem, Krull's principal ideal theorem, Cartan-Eilenberg-Bass theorem and Krull intersection theorem. The main motivation of this paper is to introduce the notions of τ q -flat modules and τ q -coheret rings, and then give some module-theoretic studies of these notions.…”

Section: Introductionmentioning

confidence: 99%

“…As our work involves q-operations, we give a brief introduction on them. For more details, refer to [24]. Recall that an ideal I of R is said to be dense if (0 : R I) := {r ∈ R | Ir = 0}, and semi-regular if there exists a finitely generated dense sub-ideal of I.…”

Section: Introductionmentioning

confidence: 99%

“…By [18, Proposition 2.2], DQ rings are exactly rings with small finitistic dimensions equal to 0. Recall from [24] that an submodule N of a Q-torsion free module M is called a q-submodule if N q ∩ M = N. If an ideal I of R is a qsubmodule of R, then I is also called a q-ideal of R. A maximal q-ideal is an ideal of R which is maximal among the q-submodules of R. The set of all maximal q-ideals is denoted by q-Max(R). q-Max(R) is exactly the set of all maximal non-semi-regular ideals of R, and thus is non-empty and a subset of Spec(R) (see [24, Proposition 2.5, Proposition 2.7]).…”

Section: Introductionmentioning

confidence: 99%

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On $τ_q$-flatness and $τ_q$-coherence

Zhang¹

2021

Preprint

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In this paper, we introduce and study the notions of τ q -flat modules and τ q -coheret rings. First, by investigating the Nagata rings of τ q -torsion theory, we show that the small finitistic dimensions of T(R[x]) are all equal to 0 for any ring R. Then, we introduce the notion of τ q -VN regular rings (i.e. over which all modules are τ q -flat), and show that a ring R is a τ q -VN regular ring if and only if T(R[x]) is a von Neumann regular ring. Finally, we obtain the Chase theorem for τ q -coheret rings: a ring R is τ q -coherent if and only if any direct product of R is τ q -flat if and only if any direct product of flat R-modules is τ q -flat. Some examples are provided to compare with the known conceptions.

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Some Characterizations of w‐Noetherian Rings and SM Rings

Zhou

Kim

Zhang

et al. 2022

Journal of Mathematics

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In this paper, we characterize w -Noetherian rings and SM rings. More precisely, in terms of the u -operation on a commutative ring R , we prove that R is w -Noetherian if and only if the direct limit of r GV -torsion-free injective R -modules is injective and that R is SM, which can be regarded as a regular w -Noetherian ring, if and only if the direct limit of GV -torsion-free (or r GV -torsion-free) reg-injective R -modules is reg-injective. As a by-product of the proof of the second statement, we also obtain that the direct and inverse limits of u -modules are both u -modules and that SM rings are regular w -coherent.

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A new semistar operation on a commutative ring and its applications

Cited by 6 publications

References 16 publications

A homological characterization of $Q_0$-Prüfer $v$-multiplication rings

A homological characterization of $Q_0$-Prüfer $v$-multiplication rings

On $τ_q$-flatness and $τ_q$-coherence

Some Characterizations of w‐Noetherian Rings and SM Rings

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