Xiaolei Zhang

On the n-perfectness under split nilpotent extensions

Wei Qi, Xiaolei Zhang

Let $π : S ↠ R$ be a split ring epimorphism with nilpotent kernel $J$ , and let $R ↪ S$ be a fixed ring-theoretic section. We prove a change-of-rings form…

Duality Between Injective Envelopes and Flat Covers over Noether Algebras

Xiaolei Zhang

It follows by Puuska that, over a commutative Noetherian ring $R$ , a morphism $i : M \to I$ of $R$ -modules is an injective envelope if and only if its Matlis dual $Hom_R(i,…

math.RA2026

Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants

Guocheng Dai, Xiaolei Zhang

In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly $ℵ_{1}$ -presented module $M$ satisfying…

The small finitistic dimensions of commutative rings, III

Xiaolei Zhang

The small finitistic dimension fPD $(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$ -modules with finite projective resolutions. In this paper, we sh…

Relative Faithful Exact Functors and Their Applications to Homological Modules

Xiaolei Zhang, Lei Qiao, Hwankoo Kim

The notions of faithfully projective, faithfully flat, and faithfully injective modules--defined as modules for which the three classical homological functors are both faithful and…

A note on Matlis localizations

Xiaolei Zhang

Let $R$ be a commutative ring and $S$ a multiplicative subset of $R$ . A ring $R$ is called an $S$ -Matlis ring if $p d_{R} R_{S} \leq 1$ . In this note, we give some new characterizations o…

math.RA2023

Little finitistic dimensions and generalized derived categories

Xiaolei Zhang, Tiwei Zhao, Dingguo Wang

In this paper, we introduced a generalization of the derived category, which is called the $n$ -derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathb…

Characterizing Baer Modules Over $τ_{q}$ -Semisimple Rings: An Extension of Baer's Problem

Xiaolei Zhang, Hwankoo Kim

In this note, we investigate the Baer splitting problem over commutative rings. In particular, we show that if a commutative ring $R$ is $τ_{q}$ -semisimple, then every Baer $R$ -modul…

On (uniformly) $S$ - $w$ -Noetherian rings and modules

Xiaolei Zhang

Let $R$ be a ring and $S$ a multiplicative subset of $R$ . We introduce and study the notions of ( $u$ -) $S$ - $w$ -Noetherian modules and ( $u$ -) $S$ - $w$ -principal ideal modules. Some cha…

math.RA2023

On the generalizations of global dimensions and singularity categories

Xiaolei Zhang, Tiwei Zhao, Dingguo Wang

For each $n \in N \cup {\infty}$ , we introduce the notion of $n$ -singularity category $D_{n - s g} (R)$ of a given ring $R$ , which can be seen as a generalization…

On $τ_{q}$ -weak global dimensions of commuative rings

Xiaolei Zhang, Ning Bian, Refat Abdelmawla Khaled Assaad +1

In this paper, the $τ_{q}$ -weak global dimension $τ_{q}$ -\cwd $(R)$ of a commutative ring $R$ is introduced. Rings with $τ_{q}$ -weak global dimension equal to $0$ are studied in terms of…

math.AC2023★ 1 cited

A note on the Cohen type theorem and the Eakin-Nagata type theorem for uniformly $S$ -Noetherian rings

Xiaolei Zhang

In this note, we give the Cohen type theorem for uniformly $S$ -Noetherian modules and the Eakin-Nagata type theorem for uniformly $S$ -Noetherian rings. We also solve an open questi…

On strongly $ϕ$ - $w$ -Flat modules and Their Homological Dimensions

Wei Qi, Xiaolei Zhang, Wei Zhao

In this paper, we introduce and study the notion of strongly $ϕ$ - $w$ -flat modules. The $ϕ$ - $w$ -weak global dimension $ϕ$ - $w$ -w.gl.dim $(R)$ of an NP-ring $R$ is also introduced and…

On $τ_{q}$ -projectivity and $τ_{q}$ -simplicity

Xiaolei Zhang

In this paper, we first introduce and study the notion of $τ_{q}$ -projective modules via strongly Lucas modules, and then investigate the $τ_{q}$ -global dimension $τ_{q}$ -\gld $(R)$ of a…

math.AC2022★ 1 cited

Strongly $ϕ$ -flat modules, strongly nonnil-injective modules and their homological dimensions

Xiaolei Zhang, Shiqi Xing, Wei Qi

In this paper, we first introduce and study the notions of strongly $ϕ$ -flat modules and strongly nonnil-injective modules. And then, we investigate the homology dimensions of modu…

math.AC2022★ 1 cited

Characterizing $S$ -Artinianness by uniformity

Xiaolei Zhang, Wei Qi

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$ . An $R$ -module $M$ is said to be a uniformly $S$ -Artinian ( $u$ - $S$ -Artinian for abbreviation) mod…

Some remarks on Lucas modules

Xiaolei Zhang, Guocheng Dai, Wei Qi

In this paper, we discuss some properties on Lucas modules. In details, we show that direct and inverse limits of Lucas modules are Lucas modules, and every $R$ -module has a Lucas…

On Cohen's theorem for Artinian modules

Xiaolei Zhang, Hwankoo Kim, Wei Qi

In this paper, we prove that a finitely embedded $R$ -module $M$ is Artinian if and only if for every prime ideal $p$ of $R$ with $(0 :_{R} M) \subseteq p$ , there e…

On two versions of Cohen's theorem for modules

Xiaolei Zhang, Hwankoo Kim, Wei Qi

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated $R$ -module $M$ is Noetherian if and only if for every prim…

Nil $_{*}$ -Noetherian rings

Xiaolei Zhang

In this paper, we say a ring $R$ is Nil $_{*}$ -Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil $_{*}$ -…