CyberSec Research

We introduce the notion of pure extending modules, a refinement of classical extending modules in which only pure submodules are required to be essential in direct summands. Fundamental properties and characterizations are established, showing that pure extending and extending modules coincide over von Neumann regular rings. As an application, we prove that pure extending modules admit decomposition patterns analogous to those in the classical theory, including a generalization of the Osofsky-Smith theorem: a cyclic module whose proper factor modules are pure extending decomposes into a finite direct sum of pure-uniform submodules. Additionally, we resolve an open problem of Dehghani and Sedaghatjoo by constructing a centrally quasi-morphic module that is not centrally morphic, arising from the link between pure-extending behavior and nonsingularity in finitely generated modules over Noetherian rings.

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra \(A\), we show that any shifted double Poisson bracket on \(A\) induces a graded Lie algebra structure on the reduced cyclic homology. Under the Kontsevich--Rosenberg principle, we further prove that the noncommutative Poisson extension is compatible with noncommutative Hamiltonian reduction. Moreover, we show that shifted double Poisson structures are independent of the choice of cofibrant resolutions and that they induce shifted Poisson structures on the derived moduli stack of representations.

In this short note, we characterise some Gorenstein versions of the concept of a group being of type $\Phi$ as introduced by Olympia Talelli. And, we also generalize a different Talelli result regarding the coincidence of the classical and the Gorenstein cohomological dimension of torsion-free groups in Kropholler's $\LH\mathscr{F}$ class.

Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group $G$, if $S$ is what we call ideally symmetrically $G$-graded, then we show that there is a bijective correspondence between the $G$-graded prime ideals of $S$ and the $G$-prime ideals of $S_e$. We use this correspondence in the case where $G$ is ordered and $S$ is ideally symmetrically $G$-graded to show that $S$ is prime if and only if $S_e$ is $G$-prime. These results generalize classical theorems by N\u{a}st\u{a}sescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically $G$-graded subrings of group rings over fully idempotent rings.

We construct a category equivalent to the category $\mathbf{Mon}$ of monoids and monoid homomorphisms, based on categories with strict factorization systems. This equivalence is then extended to the category $\mathbf{Mon_s}$ of unital semigroups and semigroup homomorphisms. By introducing suitable natural transformations, we turn these equivalences into 2-equivalences between 2-categories. The 2-category $\mathbf{Mon_s^2}$ constructed this way proves the good one to study Morita equivalence of monoids.

We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices.

We prove the existence and give a classification of toric non-commutative crepant resolutions (NCCRs) of Gorenstein toric singularities with divisor class group of rank one. We prove that they correspond bijectively to non-trivial upper sets in a certain quotient of the divisor class group equipped with a certain partial order. By this classification, we show that for such toric singularities, all toric NCCRs are connected by iterated Iyama-Wemyss mutations.

A $d$-silting object is a silting object whose derived endomorphism algebra has global dimension $d$ or less. We give an equivalent condition, which can be stated in terms of dg quivers, for silting mutations to preserve the $d$-siltingness under a mild assumption. Moreover, we show that this mild assumption is always satisfied by $\nu_d$-finite algebras. As an application, we give a counterexample to the open question by Herschend-Iyama-Oppermann: the quivers of higher hereditary algebras are acyclic. Our example is a $2$-representation tame algebra with a $2$-cycle which is derived equivalent to a toric Fano stacky surface.

The importance of studying $d$-tilting bundles, which are tilting bundles whose endomorphism algebras have global dimension $d$ (or less), on $d$-dimensional smooth projective varieties has been recognized recently. In Chan's paper, it is conjectured that a smooth projective surface has a $2$-tilting bundle if and only if it is weak del Pezzo. In this paper, we prove this conjecture. Moreover, we show that this endomorphism algebra becomes a $2$-representation infinite algebra whose 3-Calabi-Yau completion gives a non-commutative crepant resolution (NCCR) of the corresponding Du Val del Pezzo cone.

Punctual noncommutative Hilbert schemes are projective varieties parametrizing finite codimensional left ideals in noncommutative formal power series rings. We determine their motives and intersection cohomology, by constructing affine pavings and small resolutions of singularities.

We derive the canonical forms for a pair of $n\times n$ complex matrices $(E,Q)$ under transformations $(E,Q) \rightarrow (UEV,U^{-T}QV)$, and $(E,Q) \rightarrow (UEV,U^{-*}QV)$, where $U$ and $V$ are nonsingular complex matrices. We, in particular, consider the special cases of $E^TQ$ and $E^*Q$ being (skew-)symmetric and (skew-)Hermitian, respectively, that are associated with Lagrangian and Dirac subspaces and related linear-time invariant dissipative Hamiltonian descriptor systems.

For an associative algebra $A$ with a simple module $M$ with trivial endomorphisms and trivial annihilator we verify the countable separation property (CSP), i.e. we prove that there exists a list of nonzero elements $a_1, a_2,\ldots$ of $A$ such that every two-sided ideal of $A$ contains at least one such $a_i$. Based on this result we verify the countable separation property for a free associative algebra with finite or countable set of generators over any field. The countable separation property was studied before in the works of Dixmier and others but only in the context of Noetherian algebras (and a free associative algebra is very far from being Noetherian).

In this paper, we study a new class of rings, called $\sqrt{J}$-clean rings. A ring in which every element can be expressed as the addition of an idempotent and an element from $\sqrt{J(R)}$ is called a $\sqrt{J}$-clean ring. Here, $\sqrt{J(R)}=\{ z\in R : z^n\in J(R) \ \mathrm{for \ some} \ n \geq 1 \}$ where, $J(R)$ is the Jacobson radical. We provide the basic properties of $\sqrt{J}$-clean rings. We also show that the class of semiboolean and nil clean rings is a proper subclass of the class of $\sqrt{J}$-clean rings, which itself is a proper subclass of clean rings. We obtain basic properties of $\sqrt{J}$-clean rings and give a characterization of $\sqrt{J}$-clean rings: a ring $R$ is a $\sqrt{J}$-clean ring iff $R/J(R)$ is a $\sqrt{J}$-clean ring and idempotents lift modulo $J(R)$. We also prove that a ring is a uniquely clean ring if and only if it is a uniquely $\sqrt{J}$-clean ring. Finally, several matrix extensions like $T_n(R)$ and $D_n(R)$ over a $\sqrt{J}$-clean ring are explored.

We give criteria to determine when a degree-2 Azumaya algebra with $C_2$-action over a dense open subvariety of a curve extends to the entire curve as an algebra with $C_2$-action. These consist of conditions for the extension of the algebra, combined with a new condition for the extension of the algebra with the action. The new condition is testable by computer algebra systems, and we explain how the result applies to the canonical components of the character varieties of certain hyperbolic knots with order-2 symmetries. We conclude by carrying out the calculations for different symmetries of the Figure-8 knot.

An orthoset is a non-empty set $X$ together with a symmetric binary relation $\perp$ and a constant $0$ such that $x \not\perp x$ for any $x \neq 0$, and $0 \perp x$ for any $x$. Maps $f \colon X \to Y$ and $g \colon Y \to X$ between orthosets are said to form an adjoint pair if, for any $x \in X$ and $y \in Y$, $f(x) \perp g$ if and only if $x \perp g(x)$. Hilbert spaces, equipped with the usual orthogonality relation and the zero vector, provide the motivating examples of orthosets. The usual adjoints of bounded linear maps between Hilbert spaces are adjoints also in our sense. We investigate dagger categories of orthosets and maps between them, requiring that any morphism and its dagger form an adjoint pair. We indicate conditions under which such a category is unitarily dagger equivalent to the dagger category of complex Hilbert spaces and bounded linear maps.

The theory of ternary $\Gamma$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\Gamma$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $\Gamma$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $\Gamma$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $\Gamma$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $\Gamma$-algebras.

The aim of this paper is to study commuting graphs of completely $0$-simple semigroups, using the characterization of these semigroups as $0$-Rees matrix semigroups over a groups. We establish a method to decide whether the commuting graph of this semigroup construction is connected or not. If it is not connected, we also supply a way to identify the connected components of the commuting graph. We show how to obtain the diameter of the commuting graph (when it is connected) and the diameters of the connected components of the commuting graph (when it is not connected). Moreover, we obtain the clique number and girth of the commuting graph of such a semigroup, as well as two upper bounds (either of which can be the best in different situations) for its chromatic number. We also determine the knit degree of such a semigroup. Finally, we use the results regarding the properties of the commuting graph of a $0$-Rees matrix semigroup over a group to determine the set of possible values for the diameter, clique number, girth, chromatic number and knit degree of the commuting graph of a completely $0$-simple semigroup.

Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$ satisfy $A_gA_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\ldots,g_n)\in G^n$ there exist homogeneous elements $a_i\in A_{g_i}$ such that $a_1\cdots a_n\ne 0$ in $A$; furthermore, for every $g$, $h\in G$ and every $a_g\in A_g$, $a_h\in A_h$ one has $a_ga_h=\beta(g,h)a_ha_g$ for some $\beta(g,h)\in K^*$. Here $\beta(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that $\beta$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with $\beta(g,h)=\beta(g,k)$ one has $h=k$. In this paper we prove that if $G=\mathbb{Z}_2$ then every $G$-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of $\mathbb{Z}_2$-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated subalgebras of a $\mathbb{Z}_2$-graded regular algebra having a minimal regular decomposition.