arXiv subjects
Publications and source records attributed to Xiaolei Zhang.
In this note, we give several characterizations of left pure-semisimple in terms of the (pre)envelope, (pre)cover, direct limits, direct sums, inverse limits and direct products properties of pure-projective modules or pure-injective modules.
For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of $n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which can be seen as a generalization of the classical singularity category. Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements.
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 investigate the small finitistic dimensions of four types of ring constructions: polynomial rings, formal power series rings, trivial extensions and amalgamations. Besides, we show the small finitistic dimensions of a ring is less than or equal to its Krull dimension. We also give a total ring of quotients with infinite small finitistic dimension.
Let $(K,M)$ be a pair satisfying some mild condition, where $K$ is a class of $R$-modules and $M$ is a class of $R$-homomorphisms. We show that if $f:A\rightarrow B$ and $g:B\rightarrow A$ are $M$-embeddings and $A,B$ are $K_M$-injective, then $A$ is isomorphic to $B$, positively answering an question proposed by Marcos and Jiri [6].
Let $f:S\rightarrow R$ be a ring extension. We introduce and study the properties of $(R, S)_\star$-injective modules and the existences of $(R, S)_\star$-injective envelopes. Besides, we show that every $R$-module has an $(R, S)$-injective envelope when $S$ is a pure-semisimple ring.
In this note, we investigate the notion of $S$-flat preenvelopes of modules. In particular, we give an example that a ring $R$ being coherent does not imply that every $R$-module have an $S$-flat preenvelope, giving a negative answer to the question proposed by Bennis and Bouziri \cite{BB24}. Besides, we also show that $R_S$ is a coherent ring also does not imply that $R$ is an $S$-coherent ring in general.
In this paper, we introduce and study the class of $ϕ$-$w$-P-flat modules, which can be seen as generalizations of both $ϕ$-P-flat modules and $w$-P-flat modules. In particular, we obtain that the class of $ϕ$-$w$-P-flat modules is covering. We also utilize the class of $ϕ$-$w$-P-flat modules to characterize $ϕ$-von Neumann regular rings, strong $ϕ$-rings and $ϕ$-PvMRs.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we study the localization of $S$-injective modules and $u$-$S$-injective modules under $S$-Noetherian rings and $u$-$S$-Noetherian rings, respectively. The $u$-$S$-absolutely pure property is showed to be preserved under localizations over $S$-coherent rings. Besides, we give an example to show the difference between $S$-injective modules and $u$-$S$-injective modules, and some counter-examples to deny some questions proposed in \cite{B24}.
In this note, we show that a ring $R$ is $S$-coherent if and only if every finitely presented $R$-module is $S$-coherent, providing a positive answer to a question proposed in [D. Bennis, M. El Hajoui, {\it On $S$-coherence}, J. Korean Math. Soc. \textbf{55} (2018), no.6, 1499-1512]. Besides, we show that $c$-$S$-coherent rings are $S$-coherent, and give an example to show the converse is not true in general.
In this paper, we introduce and study the $q$-Krull dimension of a commutative ring via its $q$-operation. A new characterization of $τ_q$-von Neumann regular rings is obtained, and some properties of rings $q$-Krull dimension 0 are studied. Moreover, we characterize the $q$-Krull dimension of $τ_q$-Noetherian rings in terms of Krull dimension of some total quotient rings. Besides, we show that every ring with $w$-Krull dimension $0$ is a \DW\ ring, positively answers an question proposed by H. Kim.
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 ring $R$. We obtain that if $R$ is a $τ_q$-Noetherian ring, then $τ_q$-\gld$(R)=τ_q$-\gld$(R[x])=$\gld$({\rm T}(R[x]))$. Finally, we study the rings over which all modules are $τ_q$-projective (i.e., $τ_q$-semisimple rings). In particular, we show that a ring $R$ is a $τ_q$-semisimple ring if and only if ${\rm T}(R[x])$ (or ${\rm T}(R)$, or $\mathcal{Q}_0(R)$) is a semisimple ring, if and only if $R$ is a reduced ring with ${\rm Min}(R)$ finite, if and only if every reg-injective (or semireg-injective, or Lucas, or strongly Lucas) module is injective.
In this note, we study the generalized fraction properties and power series properties of $\mathcal{S}$-Noetherian rings. Actually, we investigate two questions proposed in [A. Dabbabi, A. Benhissi, Generalization of the $S$-Noetherian concept, {\it Arch. Math.} (Brno) \textbf{59}(4) (2023) 307-314.]
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) module if there is $s\in S$ such that any descending chain of submodules of $M$ is $S$-stationary with respect to $s$. $u$-$S$-Artinian modules are characterized in terms of ($S$-MIN)-conditions and $u$-$S$-cofinite properties. We call a ring $R$ is a $u$-$S$-Artinian ring if $R$ itself is a $u$-$S$-Artinian module, and then show that any $u$-$S$-semisimple ring is $u$-$S$-Artinian. It is proved that a ring $R$ is $u$-$S$-Artinian if and only if $R$ is $u$-$S$-Noetherian, the $u$-$S$-Jacobson radical ${\rm Jac}_S(R)$ of $R$ is $S$-nilpotent and $R/{\rm Jac}_S(R)$ is a $u$-$S/{\rm Jac}_S(R)$-semisimple ring. Besides, some examples are given to distinguish Artinian rings, $u$-$S$-Artinian rings and $S$-Artinian rings.
Let $R$ be a commutative ring and $S$ a multiplicative subset of $R$. A ring $R$ is called an $S$-Matlis ring if $pd_RR_S\leq 1$. In this note, we give some new characterizations of $S$-Matlis rings in terms of $S$-strongly flat modules, $S$-weakly cotorsion modules and $S$-$h$-divisible modules.
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 $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is proved to be very closely connected with its left little finitistic dimension. We also introduce and investigate the notions of $n$-exact sequences, $n$-projective (resp., $n$-injective) modules and $n$-exact complexes. In particular, we characterize the left little finitistic dimensions in terms of all above notions. Finally, we build a connection of the classical derived categories and $n$-derived categories.
Coronary artery segmentation on coronary-computed tomography angiography (CCTA) images is crucial for clinical use. Due to the expertise-required and labor-intensive annotation process, there is a growing demand for the relevant label-efficient learning algorithms. To this end, we propose partial vessels annotation (PVA) based on the challenges of coronary artery segmentation and clinical diagnostic characteristics. Further, we propose a progressive weakly supervised learning framework to achieve accurate segmentation under PVA. First, our proposed framework learns the local features of vessels to propagate the knowledge to unlabeled regions. Subsequently, it learns the global structure by utilizing the propagated knowledge, and corrects the errors introduced in the propagation process. Finally, it leverages the similarity between feature embeddings and the feature prototype to enhance testing outputs. Experiments on clinical data reveals that our proposed framework outperforms the competing methods under PVA (24.29% vessels), and achieves comparable performance in trunk continuity with the baseline model using full annotation (100% vessels).
In this paper, we introduce and study the notions of uniformly $S$-finitely presented modules and uniformly $S$-coherent rings (modules) which are "uniform" versions of ($c$-)$S$-finitely presented modules and ($c$-)$S$-coherent rings (modules) introduced by Bennis and Hajoui \cite{bh18}. Among the results, uniformly $S$-versions of Chase's result, Chase Theorem and Matlis Theorem are obtained.
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 homologies, direct products, polynomial extensions and amalgamations. Besides, we investigate the $τ_q$-weak global dimensions of polynomial rings.