Searcharxiv⌕ Search

arXiv subjects

Xiaolei Zhang

Publications and source records attributed to Xiaolei Zhang.

At least 73 records · Page 4Linked to original sources
arXiv2023

A note on a Cohen-type theorem for $w$-Artinian modules

In this note, we prove that a $w$-module $M$ is $w$-Artinian if and only if it is $w$-cofinitely generated and for every prime $w$-ideal $\mathfrak{p}$ of $R$ with $(0:_RM)\subseteq \mathfrak{p}$, there exists a $w$-submodule $N^\mathfrak{p}$ of $M$ such that $(M/N^\mathfrak{p})_w$ is $w$-cofinitely generated and $(M[\mathfrak{p}])_w\subseteq N^\mathfrak{p} \subseteq (0:_M\mathfrak{p})$, where $M[\mathfrak{p}]=\bigcap\limits_{s\in R \setminus \mathfrak{p}}s(0:_M\mathfrak{p}).$ Besides, we show that the $w$-operations are semi-star operations rather than star operations in general.

math.AC↗
arXiv2023

The small finitistic dimensions of commutative rings

Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $\fPD(R)\leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\}$ where $\grade(\m,R)$ is the grade of $\m$ on $R$ . We also show that a ring $R$ satisfies $\fPD(R)\leq 1$ if and only if $R$ is a $\DW$ ring. As applications, we show that the small finitistic dimensions of strong \Prufer\ rings and $\LPVD$s are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain examples of total rings of quotients $R$ with $\fPD(R)=n$.

math.AC↗
arXiv2023

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

Let $R$ be a commutative ring. An $R$-module $M$ is called a semi-regular $w$-flat module if $\Tor_1^R(R/I,M)$ is $\GV$-torsion for any finitely generated semi-regular ideal $I$. In this article, we show that the class of semi-regular $w$-flat modules is a covering class. Utilizing these notions, we give some homological characterizations of $\WQ$-rings and $Q_0$-\PvMR s.

math.AC↗
arXiv2023

Some remarks on Lucas modules

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 envelope and a Lucas cover. Moreover, some properties of direct and inverse limits of Lucas modules and some constructions and the unique mapping properties of Lucas envelopes and Lucas covers are investigated.

math.AC↗
arXiv2023

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

In this paper, we introduce and study the class of $ϕ$-$w$-flat modules which are generalizations of both $ϕ$-flat modules and $w$-flat modules. The $ϕ$-$w$-weak global dimension $ϕ$-$w$-w.gl.dim$(R)$ of a strongly $ϕ$-ring $R$ is also introduced and studied. We show that, for a strongly $ϕ$-ring $R$, $ϕ$-$w$-w.gl.dim$(R)=0$ if and only if $w$-$dim(R)=0$ if and only if $R$ is a $ϕ$-von Neumann ring. It is also proved that, for a strongly $ϕ$-ring $R$, $ϕ$-$w$-w.gl.dim$(R)\leq 1$ if and only if each nonnil ideal of $R$ is $ϕ$-$w$-flat, if and only if $R$ is a $ϕ$-PvMR, if and only if $R$ is a PvMR.

math.AC↗
arXiv2023

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

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 studied. We characterize $ϕ$-\Prufer\ $v$-multiplication strongly $ϕ$-rings in terms of strongly $ϕ$-$w$-flat modules.

math.AC↗
arXiv2023

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

In this paper, the notions of nonnil-injective modules and nonnil-FP-injective 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 $ϕ$-\Prufer\ rings in terms of $ϕ$-flat modules, nonnil-injective modules and nonnil-FP-injective modules.

math.AC↗
arXiv2022

On $τ_q$-flatness and $τ_q$-coherence

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.

math.AC↗
arXiv2022

Characterizing $S$-projective modules and $S$-semisimple rings by uniformity

Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\rightarrow \mathrm{Hom}_R(P,A)\rightarrow \mathrm{Hom}_R(P,B)\rightarrow \mathrm{Hom}_R(P,C)\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.

math.AC↗
arXiv2022

Nil$_{\ast}$-Noetherian rings

In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if and only if $R[x]$ is Nil$_{\ast}$-Noetherian, if and only if $R[[x]]$ is Nil$_{\ast}$-Noetherian. Then we discuss some Nil$_{\ast}$-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil$_{\ast}$-Noetherian rings in terms of Nil$_{\ast}$-injective modules and Nil$_{\ast}$-FP-injective modules. Besides, some examples are given to distinguish Nil$_{\ast}$-Noetherian rings, Nil$_{\ast}$-coherent rings and so on.

math.AC↗
arXiv2022

On two versions of Cohen's theorem for modules

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 prime ideal $\mathfrak{p}$ of $R$ with Ann$(M)\subseteq \mathfrak{p}$, there exists a finitely generated submodule $N^\mathfrak{p}$ of $M$ such that $\mathfrak{p} M\subseteq N^\mathfrak{p}\subseteq M(\mathfrak{p})$, where $M(\mathfrak{p})=\{x\in M\mid sx\in \mathfrak{p} M $ for some $s\in R \setminus \mathfrak{p} \}$. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to those for $S$-Noetherian modules and $w$-Noetherian modules.

math.AC↗
arXiv2022

On Cohen's theorem for Artinian modules

In this paper, we prove that a finitely embedded $R$-module $M$ is Artinian if and only if for every prime ideal $\mathfrak{p}$ of $R$ with $(0:_RM)\subseteq \mathfrak{p}$, there exists a submodule $N^\mathfrak{p}$ of $M$ such that $M/N^\mathfrak{p}$ is finitely embedded and $M[\mathfrak{p}]\subseteq N^\mathfrak{p}\subseteq (0:_M\mathfrak{p})$.

math.AC↗
arXiv2022

On uniformly $S$-absolutely pure modules

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $S$-pure $S$-exact sequences and $S$-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then we characterize $S$-von Neumann regular rings and uniformly $S$-Noetherian rings using $S$-absolutely pure modules.

math.AC↗
arXiv2022

Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity

Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called $u$-$S$-torsion ($u$- always abbreviates uniformly) provided that $sT=0$ for some $s\in S$. The notion of $u$-$S$-exact sequences is also introduced from the viewpoint of uniformity. An $R$-module $F$ is called $u$-$S$-flat provided that the induced sequence $0\rightarrow A\otimes_RF\rightarrow B\otimes_RF\rightarrow C\otimes_RF\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. A ring $R$ is called $u$-$S$-von Neumann regular provided there exists an element $s\in S$ satisfying that for any $a\in R$ there exists $r\in R$ such that $sa=ra^2$. We obtain that a ring $R$ is a $u$-$S$-von Neumann regular ring if and only if any $R$-module is $u$-$S$-flat. Several properties of $u$-$S$-flat modules and $u$-$S$-von Neumann regular rings are obtained.

math.AC↗
arXiv2021

New Characterizations of $S$-coherent rings

In this paper, we introduce and study the class $S$-$\mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$\mathcal{F}$-ML is closed under submodules. As an application, we obtain the $S$-version of Chase Theorem: a ring $R$ is $S$-coherent if and only if any direct product of $R$ is $S$-flat if and only if any direct product of flat $R$-modules is $S$-flat. Consequently, we provide an answer to the open question proposed by D. Bennis and M. El Hajoui [3].

math.AC↗
arXiv2021

Semi-regular flat modules over strong Prüfer rings

We first introduce and study the notion of semi-regular flat modules, and then show that a ring $R$ is a strong \Prufer\ ring if and only if every submodule of a semi-regular flat $R$-module is semi-regular flat, if and only if every ideal of $R$ is semi-regular flat, if and only if every $R$-module has a surjective semi-regular flat (pre)envelope.

math.AC↗