arXiv subjects

Fanggui Wang

Publications and source records attributed to Fanggui Wang.

6 recordsLinked to original sources

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↗

arXiv2022

Uniformly $S$-Noetherian rings

Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated sub-ideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for $u$-$S$-Noetherian rings. Besides, the $u$-$S$-Noetherian properties on several ring constructions are given. The notion of $u$-$S$-injective modules is also introduced and studied. Finally, we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetherian rings.

math.AC↗

arXiv2021

Metamagnetic Transitions in Few-Layer CrOCl Controlled by Magnetic Anisotropy Flipping

The pivotal role of magnetic anisotropy in stabilising two-dimensional (2D) magnetism has been widely accepted, however, direct correlation between magnetic anisotropy and long-range magnetic ordering in the 2D limit is yet to be explored. Here, using angle- and temperature-dependent tunnelling magnetoresistance, we report unprecedented metamagnetic phase transitions in atomically-thin CrOCl, triggered by magnetic easy-axis flipping instead of the conventional spin flop mechanism. Few-layer CrOCl tunnelling devices of various thicknesses consistently show an in-plane antiferromagnetic (AFM) ground state with the easy axis aligned along the Cr-O-Cr direction (b-axis). Strikingly, with the presence of a magnetic field perpendicular to the easy-axis (H||c), magnetization of CrOCl does not follow the prevalent spin rotation and saturation pattern, but rather exhibits an easy-axis flipping from the in-plane to out-of-plane directions. Such magnetic anisotropy controlled metamagnetic phase transitions are manifested by a drastic upturn in tun- nelling current, which shows anomalous shifts towards higher H when temperature increases. By 2D mapping of tunnelling currents as a function of both temperature and H, we determine a unique ferrimagnetic state with a superstructure periodicity of five unit cells after the field-induced metam- agnetic transitions. The feasibility to control 2D magnetism by manipulating magnetic anisotropy may open enormous opportunities in spin-based device applications.

cond-mat.str-el↗

arXiv2020

A new version of a theorem of Kaplansky

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the $w$-version of this theorem, where $w$ is a hereditary torsion theory for modules over a commutative ring.

math.AC↗

arXiv2015

On $L_{n}$-Injective Modules and $L_{n}$-Injective Dimensions

Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimension $\leq n$, perfect rings, and $L_n$-hereditary rings, more properties and applications of $L_{n}$-injective modules, $L_{n}$-injective dimensions of modules and $\mathcal{L}_{n}$-global dimensions of rings are given.

math.RA↗

arXiv2015

Super finitely presented modules and Gorenstein projective modules

Let $R$ be a commutative ring. An $R$-module $M$ is said to be super finitely presented if there is an exact sequence of $R$-modules $\cdots\rightarrow P_n\rightarrow\cdots \rightarrow P_1\rightarrow P_0\rightarrow M\rightarrow 0$ where each $P_i$ is finitely generated projective. In this paper it is shown that if $R$ has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if $R$ has the property (C) that every super finitely presented module has finite projective dimension, then $R$ is $K_0$-regular, i.e., $K_0(R[x_1,\cdots,x_n])\cong K_0(R)$ for all $n\geq 1$.

math.AC↗