arXiv subjects

Xiaolei Zhang

Publications and source records attributed to Xiaolei Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Goldman primes and higher cyclic presentations for the small finitistic dimension

Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\sup\{depth R_{\mathfrak p}\mid \mathfrak p\in GSpec(R)\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\geq2$ and $h\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.

math.AC↗

arXiv2026

On the iso-Artinianness of one-dimensional Noetherian rings

Let $R$ be a one-dimensional commutative Noetherian ring with a unique minimal prime $\mathfrak p$ such that $D=R/\mathfrak p$ is a principal ideal domain. We give a complete characterization of the iso-Artinian property in this class, that is, the following conditions are equivalent: $R$ is iso-Artinian; $\mathfrak pR_{\mathfrak p}=0$; $len_R(\mathfrak p)<\infty$; $\mathfrak p/\mathfrak p^2$ is torsion over $D$. As a consequence, we completely answer the Question~3.9 of Daneshvar and Divaani-Aazar: under their hypotheses $Min R\subsetneq Ass R$, the ring is iso-Artinian precisely when its nilradical has finite length, and it is non-iso-Artinian precisely when the conormal module has positive rank.

math.AC↗

arXiv2026

Baer Splitting Beyond $τ_q$-Semisimplicity

For a commutative ring $R$, we call an $R$-module $B$ Baer if $\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $τ_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes $τ_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\varepsilon]/(\varepsilon^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not $τ_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\EC(D)$-module is projective, whereas $\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not $τ_q$-semisimple.

math.AC↗

arXiv2026

Characterizing categoricity in the class $Add(M)$

We study categoricity of the additive closure $\operatorname{Add}(M)$, consisting of all direct summands of arbitrary direct sums of copies of a fixed module $M$. For an $η$-generated module $M$, we prove that categoricity in a single cardinal $λ\ge θ= \max\{|R|,η\}$ is equivalent to the stabilization condition $P^{(η)} \cong M^{(η)}$ for every nonzero $η$-generated $P \in \operatorname{Add}(M)$, or equivalently $\operatorname{Add}(P)=\operatorname{Add}(M)$. This condition implies that every $X \in \operatorname{Add}(M)$ of cardinality $λ> ν_M = \max\{θ, |\mathcal{S}_η(M)|\}$ is isomorphic to $M^{(λ)}$, where $\mathcal{S}_η(M)$ is a representative set of the nonzero $η$-generated modules in $\operatorname{Add}(M)$. Since $|\mathcal{S}_η(M)| \le 2^θ$, we obtain a uniform ZFC tail above $2^θ$. For projective modules, the criterion reduces to Bass's right $p$-connectedness. For pure-projective modules, it forces von Neumann regularity and simplicity: the pure-projective modules are eventually categorical if and only if $R$ is a simple von Neumann regular ring. In the commutative case, this recovers precisely the fields. Our approach combines Walker's decomposition theorem, support reduction, and Eilenberg absorption, and we highlight several subtle pitfalls in naive downward transfer arguments.

math.RA↗

arXiv2026

Duality Between Injective Envelopes and Flat Covers over Noether Algebras

It follows by Puuska that, over a commutative Noetherian ring $R$, a morphism $i : M \rightarrow I$ of $R$-modules is an injective envelope if and only if its Matlis dual $Hom_R(i, E)$ is a flat cover for some injective cogenerator $E$, and equivalently for every injective $R$-module $E$. In this paper, we will generalize this result to non-commutative Noetherian algebras.

math.AC↗

arXiv2026

On S-(h-)divisible modules and their S-strongly flat covers

It was proved in [3] that every h-divisible modules admits an strongly flat cover over all integral domains; and every divisible module over an integral domain R admits a strongly flat cover if and only if R is a Matlis domain. In this paper, we extend these two results to commutative rings with multiplicative subsets.

math.AC↗

arXiv2026

Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants

In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly $\aleph_1$-presented module $M$ satisfying Ext$_R^{\ge 1}(M,D^{(\aleph_1)})=0$ belongs to the left orthogonal class of $\overline{D}$, where $\overline{D}$ is the definable of $D$. This yields the consequence that every $\aleph_1$-generated strongly Gorenstein projective module is Gorenstein flat. Furthermore, we investigate when a flat Gorenstein projective module is projective. Finally, for any two-sided $\aleph_1$-coherent ring $R$, we prove the identity $\operatorname{silp}R+\operatorname{silp}R^{\mathrm{op}}=\operatorname{spli}R+\operatorname{spli}R^{\mathrm{op}}$.

math.RA↗

arXiv2026

What makes a useful molecular model of biochar? A community roadmap

Biochars are disordered carbonaceous materials produced by biomass pyrolysis, with applications spanning soil amendment, water remediation, carbon storage, and functional materials. Although they share structural features with other disordered carbons such as coal, kerogen, and activated carbons, the questions posed to biochar models are distinct, and no single model can answer all of them equally well. Model usefulness must be defined relative to a specific question and validated against independent experimental observables. This community roadmap, arising from a CECAM workshop, critically maps current molecular approaches: experimentally guided top-down reconstruction, mimetic bottom-up simulation, and hybrid methods. We argue that first-generation models have been more successful than is often acknowledged, provided they are built at sufficient length scale and with explicit control over microporosity and bulk chemistry. Structural and equilibrium interfacial properties are increasingly tractable with classical force fields, whereas dynamic and reactive behaviours require selective use of reactive methods within multiscale workflows. A parallel, largely unaddressed gap concerns the mineral and ash components of biochar, and the changes the material undergoes during ageing in soil. We identify seven open questions current models cannot yet answer reliably, and five community priorities: force field benchmarking, open model and data repositories, shared classification and metadata standards, ensemble validation, and training in reproducible practice. Across these, sustained interaction with experimentalists is essential to ground models in real observables and document where they fail. Progress will be accelerated by adapting transferable methods from coal, kerogen, and clay-organic matter frameworks rather than repeating trial-and-error development.

cond-mat.mtrl-sci↗

arXiv2026

A note on the $S$-version of Noetherianity

It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar results on $S$-Noetherian rings and $S$-$\ast_w$-PIDs, where $S$ is a multiplicative subset and $\ast$ is a star operation. In particular, we gave negative answers to the open questions proposed by Hamed and Hizem \cite{hh16}, Kim and Lim \cite{kl18}, and Lim \cite{l18} in terms of valuation domains, respectively.

math.AC↗

arXiv2026

Unison: Harmonizing Motion, Speech, and Sound for Human-Centric Audio-Video Generation

Motion, speech, and sound effects are fundamental elements of human-centric videos, yet their heterogeneous temporal characteristics make joint generation highly challenging. Existing audio-video generation models often fail to maintain consistent alignment across these modalities, leading to noticeable mismatches between motion, speech, and environmental sounds. We present Unison, a unified framework that explicitly promotes coherence across the motion, speech, and sound modalities. Within the audio stream, Unison employs a semantic-guided harmonization strategy that decouples the generation of speech and sound-effect components. Leveraging bidirectional audio cross-attention and semantic-conditioned gating for semantic-driven adaptive recomposition, this approach effectively mitigates speech dominance and enhances acoustic clarity. For audio-motion synchronization, we propose a bidirectional cross-modal forcing strategy where the cleaner modality guides the noisier one through decoupled denoising schedules, reinforced by a progressive stabilization strategy. Extensive experiments demonstrate that Unison achieves state-of-the-art performance in both audio perceptual quality and cross-modal synchronization, highlighting the importance of explicit multimodal harmonization in human-centric video generation.

cs.CV↗

arXiv2026

LITMUS: Benchmarking Behavioral Jailbreaks of LLM Agents in Real OS Environments

The rapid proliferation of LLM-based autonomous agents in real operating system environments introduces a new category of safety risk beyond content safety: behavior jailbreak, where an adversary induces an agent to execute dangerous OS-level operations with irreversible consequences. Existing benchmarks either evaluate safety at the semantic layer alone, missing physical-layer harms, or fail to isolate test cases, letting earlier runs contaminate later ones. We present LITMUS (LLM-agents In-OS Testing for Measuring Unsafe Subversion), a benchmark addressing both gaps via a semantic-physical dual verification mechanism and OS-level state rollback. LITMUS comprises 819 high-risk test cases organized into one harmful seed subset and six attack-extended subsets covering three adversarial paradigms (jailbreak speaking, skill injection, and entity wrapping), plus a fully automated multi-agent evaluation framework judging behavior at both conversational and OS-level physical layers. Evaluation across frontier agents reveals three findings: (1) current agents lack effective safety awareness, with strong models (e.g., Claude Sonnet 4.6) still executing 40.64% of high-risk operations; (2) agents exhibit pervasive Execution Hallucination (EH), verbally refusing a request while the dangerous operation has already completed at the system level, invisible to every prior semantic-only framework; and (3) skill injection and entity wrapping attacks achieve high success rates, exposing pronounced agent vulnerabilities. LITMUS provides the first standardized platform for reproducible, physically grounded behavioral safety evaluation of LLM agents in real OS environments.

cs.CR↗

arXiv2026

From Thinker to Society: Security in Hierarchical Autonomy Evolution of AI Agents

Artificial Intelligence (AI) agents have evolved from passive predictive tools into active entities capable of autonomous decision-making and environmental interaction, driven by the reasoning capabilities of Large Language Models (LLMs). However, this evolution has introduced critical security vulnerabilities that existing frameworks fail to address. The Hierarchical Autonomy Evolution (HAE) framework organizes agent security into three tiers: Cognitive Autonomy (L1) targets internal reasoning integrity; Execution Autonomy (L2) covers tool-mediated environmental interaction; Collective Autonomy (L3) addresses systemic risks in multi-agent ecosystems. We present a taxonomy of threats spanning cognitive manipulation, physical environment disruption, and multi-agent systemic failures, and evaluate existing defenses while identifying key research gaps. The findings aim to guide the development of multilayered, autonomy-aware defense architectures for trustworthy AI agent systems.

cs.CR↗

arXiv2026

The small finitistic dimensions of commutative rings, III

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 show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type.

math.AC↗

arXiv2026

Relative Faithful Exact Functors and Their Applications to Homological Modules

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 exact--play fundamental roles across various areas of algebra. In this paper, we extend these notions to the setting of $w$-operation theory. By introducing the concept of $w$-faithfully exact functors, we define and investigate the notions of $w$-faithfully projective, $w$-faithfully flat, and $w$-faithfully injective modules. We establish their fundamental properties and demonstrate their effectiveness in generalizing classical results.

math.AC↗

arXiv2026

Almost Noetherian rings and modules

In this paper, we investigate the notions of almost Noetherian rings and modules. In details, we give the Cohen type theorem, Eakin-Nagata type theorem, Kaplansky type Theorem and Hilbert basis theorem and some other rings constructions for almost Noetherian rings. In particular, we resolve a question proposed in \cite[9, B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025] under a certain condition.

math.AC↗

arXiv2026

A survey on the uniform $S$-version of rings, modules and their homological theories

This survey provides a comprehensive overview of the recent advancements in the theory of ``uniformly $S$''-algebraic structures in commutative ring theory. Originating from the classical concepts of Noetherian, coherent, von Neumann regular, and semisimple rings, the introduction of a multiplicative subset $S$ has led to the development of $S$-Noetherian, $S$-coherent, and other $S$-analogues. However, the element $s \in S$ in the original definitions often depends on the ideal or module under consideration. To overcome this limitation and enable deeper module-theoretic characterizations, the notion of "uniformly $S$" (abbreviated as $u$-$S$) was introduced. This survey systematically presents the definitions, characterizations, and properties of $u$-$S$-torsion modules, $u$-$S$-exact sequences, and the subsequent uniform analogues of fundamental module classes: $u$-$S$-finitely presented, $u$-$S$-Noetherian, $u$-$S$-coherent, $u$-$S$-flat, $u$-$S$-projective, $u$-$S$-injective, and $u$-$S$-absolutely pure modules. We then explore the associated uniform homological dimensions, including the $u$-$S$-weak global dimension, the $u$-$S$-global dimension, and their interplay with polynomial rings and localizations. The survey also covers structural ring classes such as $u$-$S$-von Neumann regular, $u$-$S$-semisimple, $u$-$S$-Artinian, $u$-$S$-multiplication rings, and rings with $u$-$S$-Noetherian spectrum.

math.AC↗

arXiv2026

CODE: A Contradiction-Based Deliberation Extension Framework for Overthinking Attacks on Retrieval-Augmented Generation

Introducing reasoning models into Retrieval-Augmented Generation (RAG) systems enhances task performance through step-by-step reasoning, logical consistency, and multi-step self-verification. However, recent studies have shown that reasoning models suffer from overthinking attacks, where models are tricked to generate unnecessarily high number of reasoning tokens. In this paper, we reveal that such overthinking risk can be inherited by RAG systems equipped with reasoning models, by proposing an end-to-end attack framework named Contradiction-Based Deliberation Extension (CODE). Specifically, CODE develops a multi-agent architecture to construct poisoning samples that are injected into the knowledge base. These samples 1) are highly correlated with the use query, such that can be retrieved as inputs to the reasoning model; and 2) contain contradiction between the logical and evidence layers that cause models to overthink, and are optimized to exhibit highly diverse styles. Moreover, the inference overhead of CODE is extremely difficult to detect, as no modification is needed on the user query, and the task accuracy remain unaffected. Extensive experiments on two datasets across five commercial reasoning models demonstrate that the proposed attack causes a 5.32x-24.72x increase in reasoning token consumption, without degrading task performance. Finally, we also discuss and evaluate potential countermeasures to mitigate overthinking risks.

cs.CR↗

arXiv2026

Almost coherent rings

Inspired from the work of P. Scholze on the finiteness of $\mathbf{F}_{p}$-cohomology groups of proper rigid-analytic varieties over $p$-adic fields, Zavyalov recently introduced the notion of almost coherent rings, which plays a key role in the almost ring theory. In this paper, we characterize almost coherent rings in terms of almost flat modules and almost absolutely pure modules, integrating numerous classical results into almost mathematics. Besides, we show that every almost coherent $R$-module is not almost isomorphic to a coherent $R$-module, giving a negative answer to a question proposed in [14,B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025].

math.AC↗