arXiv subjects

Xiaolei Zhang

Publications and source records attributed to Xiaolei Zhang.

At least 19 recordsLinked to original sources

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

Quasi-projective dimensions over homological ring epimorphisms

In this paper, we study quasi-projective dimensions of modules over rings connected by homological ring epimorphisms. We establish several inequalities relating the quasi-projective dimensions over the source and target rings. As applications, we investigate the quasi-global dimension of triangular matrix rings and provide examples showing that the class of modules with finite quasi-projective dimension does not satisfy the 2-out-of-3 property in general.

math.RA↗

arXiv2026

Von Neumann regular and semisimple rings realized as principal quotients of domains

In this paper, we show that every commutative von Neumann regular ring is a quotient of a characteristic-zero Bézout domain by its principal Jacobson radical. We then characterize the commutative semisimple rings that occur as quotients of Dedekind domains. For a finite product of fields $R=\prod_{i=1}^n k_i$, such a presentation exists exactly when the cardinals $|k_i|^{\aleph_0}$ are all equal. When this condition holds, the source may be taken to be a characteristic-zero semilocal principal ideal domain, and the quotient ideal may again be its Jacobson radical.

math.AC↗

arXiv2026

The silp--spli equality for group rings

Let $R$ be an arbitrary associative ring with an identity. In this paper, we show that the equality $\silp(R)=\spli(R)$ implies $\silp(RG)=\spli(RG)$ for any group $G$.

math.RA↗

arXiv2026

Every commutative perfect ring is a quotient of an almost perfect domain

In this paper, we show that every nonzero commutative perfect ring is a quotient of a one-dimensional semilocal almost perfect domain of characteristic zero. The domain can be chosen to have as many maximal ideals as the given ring has local factors. The construction uses coefficient discrete valuation rings, triangular presentations of purely inseparable residue extensions, and approximation for finitely many independent rank-one valuations. Independent error variables keep the construction inside a field. A normal-form argument shows that the divided elements used to get perfect principal quotients do not kill any further good monomial. In positive residue characteristic, the choice $a=2c+1$ with $c\ge n$ protects the nonzero powers of $p$ in a target ring of characteristic $p^n$. We also give local and mixed-characteristic consequences, and examples with non-nilpotent radicals.

math.AC↗

arXiv2026

On the square of an isosimple module

Let $S$ be an isosimple module, that is, a nonzero module isomorphic to every nonzero submodule of itself. In this paper, we investigate if the square $S^2 := S \oplus S$ is isoantian when $S$ is an isosimple module. If the base ring is commutative, then the answer is yes. However, for noncommutative rings this is no longer true even when $\mathrm{End}_R(S)$ is a discrete valuation ring. We also give some positive results coming from quasi-projectivity and Morita equivalence. Finally, we consider when every finite powers of an isosimple module are isoantian provided it holds for the square. Our related result says that if $S^2$ is isoantian and $E = \mathrm{End}_R(S)$ is either commutative Noetherian or right semihereditary, then every finite power of $S$ is isoantian.

math.RA↗

arXiv2026

Finitistic dimensions of formal power series rings

We consider the big finitistic projective and flat dimensions of a commutative ring and of its formal power series ring. When the ring is Noetherian, both invariants increase by one. Coherence of the power series ring does not force either equality to hold. For every $n\ge0$ and every $d$ with $n+1\le d\le\infty$, the flat pair $(n,d)$ can be realized. The finite projective pairs are realized when $n\ge1$, and also for $(0,1)$. For every $n\ge0$, one family realizes $(n,\infty)$ for both invariants at the same time. The seed is a zero-dimensional perfect local ring. Its power series ring has countably presented modules whose flat dimension is $r$ and whose projective dimension lies between $r$ and $r+1$, for every $r\ge1$. The remaining finite projective case is controlled by a dichotomy for perfect coefficient rings. If the Jacobson radical is nilpotent, the power series value is one; if it is not nilpotent, there is a module of projective dimension exactly three. In particular, the pair $(0,2)$ is impossible. For one fixed countable perfect ring, the power series ring has finitistic flat dimension one. Its finitistic projective dimension is $m+2$ when $2^{\aleph_0}=\aleph_m$ with $1\le m<\infty$, and is infinite when $2^{\aleph_0}>\aleph_ω$.

math.AC↗

arXiv2026

On the $ϕ$-Weak Global Dimensions and $ϕ$-Global Dimensions

Let $R$ be a $ϕ$-ring and put $N=Nil(R)$. We determine the $ϕ$-weak global dimension of every $ϕ$-Prüfer ring. Its only possible values are $0$, $1$, and $\infty$: the value is $0$ when $R/N$ is a field, it is $1$ when $R$ is strong and $R/N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\toϕ(R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R/N$, and its self-Tor groups satisfy \[ Tor^R_{2n}(K,K)=0\quad(n\geq 1), \qquad Tor^R_{2n+1}(K,K)\neq 0\quad(n\geq 0). \] It follows that $fd_RK=\infty$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $ϕ$-global dimension of $ϕ$-Dedekind rings. For strong $ϕ$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\to R/N$ splits. If $D=R/N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then \[ R\cong T\times_Q D. \] Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\to D$, satisfies \[ fd_R(M)=fd_D(M), \qquad pd_R(M)=pd_D(M). \] Consequently, for every strong $ϕ$-ring, \[ϕ-wgd(R)=wgld(R/N), \qquad ϕ-gld(R)=gld(R/N). \]

math.AC↗

arXiv2026

The finitistic weak dimensions of Gaussian rings

We prove that every commutative Gaussian ring has finitistic weak dimension at most two. The proof works with a fixed module of finite flat dimension and does not assume that the weak global dimension of the ring is finite. For a local Gaussian ring $A$, with nilradical $N$, the kernel $I$ of $A\to A_N$ satisfies $NI=I^2=0$. A hypothetical module $M$ of flat dimension three gives a nonzero torsion module $H=\Tor_2^A(A/N,M)$ over the valuation domain $A/N$, with $H\cong\Tor_1^A(I,M)$ and $I\otimes_{A/N}H=0$. Two carefully controlled localizations, a propagation argument in the chain ring $A/D$, where $D=\{x:x^2=0\}$, and an annihilator factorization in a filtered system of cyclic modules yield a contradiction.

math.AC↗

arXiv2026

Two remarks on Almost Projective Modules

For a commutative ring $R$, an almost projective module is a module whose localization at every maximal ideal is free, with no restriction on rank. We prove that every almost projective $R$-module is projective if and only if $R$ is semilocal. Next, we turn to investigate when every $R$-module has an almost projective envelope. We show that under certain conditions, every $R$-module has an almost projective envelope if and only if it is coherent and locally Artinian. We also show that the conditions can not be omitted in terms of dual numbers rings over von Neumann regular rings.

math.AC↗

arXiv2026

Total Rings of Quotients with Arbitrary Weak Global Dimensions

For every nonnegative integer $n$, we construct a commutative total ring of quotients of weak global dimension exactly $n$. For $n\geq1$ the examples are reduced and non-coherent. The construction applies to any nonzero local ring $(A,\m)$: one adjoins countably many residue-field coordinates subject to eventual agreement with the residue of the $A$-coordinate. The resulting ring $\T(A)$ satisfies $Q(\T(A))=\T(A)$ and $\wgd\T(A)=\wgd A$. Its prime spectrum, maximal localizations, nilradical, Jacobson radical, and idempotents are described explicitly. For every pair of modules, positive-degree Tor over $\T(A)$ is naturally identified with Tor over $A$ after quotienting by a projective ideal generated by orthogonal idempotents. We prove that $\T(A)$ is coherent if and only if $A$ is a field. Specializing to regular local polynomial rings yields the prescribed dimensions, with explicit Koszul witnesses for the lower bounds. We also examine the minimal spectrum, the distinction between global and local Prüfer conditions, and a finitely presented cyclic module of flat dimension one and projective dimension two.

math.AC↗

arXiv2026

On the n-perfectness under split nilpotent extensions

Let $π:S\twoheadrightarrow R$ be a split ring epimorphism with nilpotent kernel $J$, and let $R\hookrightarrow S$ be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if $X$ is a left $S$-module satisfying $\Tor_i^S(R,X)=0$ for every $i>0$, then \[ \pd_S X=\pd_R(R\otimes_S X) \] in $\mathbb N\cup\{0,\infty\}$. In particular, for every flat left $S$-module $F$, \[ \pd_SF=\pd_R(F/JF), \] and $F/JF$ is flat over $R$. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over $S$ and over $R$. Consequently, left $n$-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring $R$ and an arbitrary $R$-module $M$, this applies to the idealization $S=R\ltimes M$ without any projectivity, flatness, finite generation, or finite flat-dimension assumption on $M$. We obtain a list of equivalent characterizations of the condition that $R\ltimes M$ is $n$-perfect, together with \[ \cotD(R\ltimes M)=\cotD(R). \] We further derive exact preservation of the flat-projective-dimension spectrum, strict $n$-perfectness, pointwise localization, arbitrary iterated idealizations, $n$-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.

math.AC↗

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↗