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Lattice Definability of Finite Matrix Rings

In this paper, we give necessary and sufficient conditions for a product of EP elements to be EP, in Dedekind-finite ∗ -rings. The proofs rely on descriptions of Morita contexts (via Peirce matrix forms) as corner rings of 2 × 2 matrix rings.

In this paper, we introduce a new class of rings calling them 2-UNJ rings, which generalize the well-known 2-UJ, 2-UU and UNJ rings. Specifically, a ring [Formula: see text] is called 2-UNJ if, for every unit [Formula: see text] of [Formula: see text], the inclusion [Formula: see text] holds, where [Formula: see text] is the set of nilpotent elements and [Formula: see text] is the Jacobson radical of [Formula: see text]. We show that every 2-UJ, 2-UU or UNJ ring is 2-UNJ, but the converse does not necessarily hold, and we also provide counter-examples to demonstrate this explicitly. We, moreover, investigate the connections between these rings and other algebraic properties such as being potent, tripotent, regular and exchange rings, respectively. In particular, we thoroughly study some natural extensions, like matrix rings and Morita contexts, obtaining new characterizations that were not addressed in previous works. Furthermore, we establish conditions under which group rings satisfy the 2-UNJ property. These results not only provide a better understanding of the structure of 2-UNJ rings, but also pave the way for future intensive research in this area. In addition, our achievements considerably improve on recent results given by Tin, On 2-UNJ rings, Asian-European J. Math. 19 (2026) 2550105.

Let [Formula: see text] be an algebra over a unital commutative ring [Formula: see text], and let [Formula: see text] be an algebraic homomorphism. In this article, we consider a linear map [Formula: see text] that satisfies one of the following conditions: [Formula: see text] [Formula: see text] or [Formula: see text] where [Formula: see text] is the Lie product in [Formula: see text]. We characterize linear maps [Formula: see text] under automorphisms [Formula: see text] by reducing to the case [Formula: see text] (the identity map on [Formula: see text]). Using these equivalences, we further characterize the linear maps [Formula: see text] on generalized matrix algebras; triangular algebras; von Neumann algebras; standard operator algebras and nest algebras for every automorphism [Formula: see text] on these algebras. We also obtain similar results for Lie [Formula: see text]-centralizers. Some of our results generalize some of the previous results

Separation is one of the most important processes in all chemical industries, including the petroleum industry. Ethane gas separation is the simplest example of a gas mixture separation process. A mathematical model and dynamic simulation of the ethane gas separation unit at a natural gas liquefaction (NGL) plant were developed, focusing on developing an accurate mathematical representation that relies solely on temperature as the main dynamic variable, without introducing flow rate or pressure effects. This allows for a flexible and realistic simulation of the system's behaviour. The mathematical model was built using first-order differential equations that describe the temporal relationship between the temperatures at the inlets and outlets of the system components using matrix algebra. Actual operational data recorded from the work site were used, providing the model with a high degree of reliability and validity in simulating the system's actual performance under both steady-state and unsteady-state operating conditions. The simulation was implemented using the MATLAB/Simulink environment to represent the thermal relationships and analyze the dynamic interaction between the system units. The model results demonstrate good agreement with practical applications, making it possible to provide a ready-to-use simulation as a testbed for advanced process monitoring or operational decision support. Through system simulation, the model's ability to accurately and realistically represent system behaviour was demonstrated, matching the actual operational behavior with the help of gain factors. This work represents a qualitative contribution to the field of modelling industrial thermal systems and paves the way for further development toward building more comprehensive models for gas processing units.

The Satisfiability Problem (SAT), a fundamental NP-complete problem, is widely applied in integrated circuit verification, artificial intelligence planning, and other fields, where the growing scale and complexity of practical problems demand higher solving efficiency. Due to redundant search paths, serialized reasoning steps, and inefficient pure literal detection, traditional serial SAT solvers require efficient parallelization of the pure literal rule. This paper adopts a parallel solving algorithm for the pure literal rule based on matrix representation. The algorithm can solve the shortcomings of poor universality, insufficient parallel collaborative mechanisms, and clause reduction. We first introduce a Clause-Numerical Incidence Matrix (CNIM) representation to provide a unified mathematical model for parallel operations. Second, we design a Column Vectors Pure Literal Parallel Topological Detection (CVPLPTD) algorithm that achieves pure literal detection with O(mn/p) time complexity (p being the number of parallel threads) within the coefficient range [1.0×mn/p, 1.2×mn/p]. Finally, we adopt a dynamic matrix reduction strategy that compresses the matrix scale through row and column deletion after each pure literal assignment to reduce computational load. These innovations integrate matrix algebra and parallel computing, effectively breaking through the efficiency limitations of solving large-scale SAT problems while ensuring good universality across different computing platforms.

Let T = ( A 0 U B ) be a formal triangular matrix ring, where A and B are rings and U is a ( B , A ) -bimodule. Firstly, the strongly FP-injective T-modules are characterized provided that the left B-module U is finitely presented. Then the Gorenstein strongly FP-injective T-modules are investigated under some mild conditions. Lastly, we use Gorenstein strongly FP-injective modules developed previously to establish recollements. These abundant theory indicate that the Gorenstein strongly FP-injective modules over non-coherent rings share many elegant properties as Ding injective modules, whose performance usually needs the coherent condition on the ring.

Abstract A nonzero element a a is called 1-Sylvester in a ring R R , if there exist b , c ∈ R b,c\in R such that 1 = a b + c a 1=ab+ca . In this article, we study such elements, mainly in matrix rings over commutative rings. In particular, we study the case when b = c b=c , when b b is called an anticommutator inverse for a a .

On Tsallis relative entropies and their preservers on positive cones in operator algebras and in matrix algebras

ABSTRACT This paper establishes several equivalent conditions for a unique positive definite solution of the unified algebraic Lyapunov matrix equation (UALE). Then, two iterative algorithms for the UALE driven by error precision are proposed, which can be accelerated by optimizing the calculation of series terms. The algorithm's convergence, computational complexity, and optimal parameter selection are discussed, supported by a series of numerical experiments that illustrate the effectiveness and superiority of our results. Finally, by constructing a positive semi‐definite matrix and employing matrix‐singular inequality, a new robust bound for the unified perturbed system is derived, generalizing some of the existing results. Some numerical experiments demonstrate its practicality and advantage.

On complex hybrid numbers: Algebraic structures, matrix representations, and geometric interpretations

Irredundant generating sets for matrix algebras

Rota–Baxter operators of weight zero on the matrix algebra of order three without unit in kernel

Characterizations of the canonical trace on full matrix algebras

Embedding matrix algebras into ultragraph Leavitt path algebras and applications

Abstract Let be a simple‐artinian ring with involution. This means that is isomorphic to a matrix ring over a ring that is either a skew field or the direct sum of a skew field and its opposite, and is given in terms of an involution of . We show that an arbitrary pseudo‐quadratic module defined over can be obtained by a tensor product construction from a pseudo‐quadratic module defined over and we apply this result to give a uniform description of arbitrary pseudo‐maximal parabolic subgroups of arbitrary classical groups in terms of pseudo‐quadratic modules.

We investigate the clique numbers and structural properties of commuting graphs associated with direct sum matrix rings over finite commutative rings. For a finite commutative ring L with unity, we study the commuting graph \(\Gamma(M(m \oplus m, L))\) whose vertex set consists of all non-central matrices in \(M(m \oplus m, L)\), where two distinct vertices are adjacent if and only if they commute. Our main contributions establish fundamental lower bounds for the clique number \(\omega\Gamma(M(m \oplus m, L)))\) across various ring structures. We prove that for any finite commutative ring R with unity and positive integer \(m \geq 3\), the clique number satisfies \(\omega(\Gamma(M(m, R))) \geq |R|^{2m} - |R|^2\). For rings isomorphic to \({Z}_{p^r}\) where \(r \geq 3\) is odd, we establish the improved bound \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^{r-1})^{m^2-m}(p^{r+1})^{m-1}p^{2r} - p^{2r}\}\). When \(r \geq 2\) is even, the bound becomes \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^r)^{m^2-1}p^{2r} - p^{2r}\}\). Our approach combines sophisticated matrix-theoretic techniques with graph-theoretic analysis to construct explicit maximal cliques and derive optimal bounds. The results provide new insights into the intersection of algebraic graph theory and matrix ring theory, with potential applications in coding theory and combinatorial optimization.

Quantum entanglement is a core resource for quantum information processing, and entanglement witnesses are indispensable tools for detection and verification. This paper first clarifies some already-known properties of entanglement witnesses and the Bell State. By applying mathematical tools such as Kronecker product and matrix algebra, this paper analyzes elements in the Hermitian matrix and discusses the conditions for it to form an entanglement witness. This paper also considers both necessary and sufficient conditions of entanglement witness to impose restrictions on elements and investigated several inequality properties of two-qubit entanglement witness to detect Bell states.

Abstract Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$ , and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$ . To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$ , we recover the conditions given in [7] for the existence of a unital partial action of $G$ on $R$ .