Search Results (128)

We characterize all additive biderivations on the incidence algebra $I(P,\mathcal{R})$ of a locally finite poset P over a commutative ring with unity $\mathcal{R}$. By decomposing P into its connected chains, we prove that any additive biderivation splits uniquely into a sum of inner biderivations and extremal ones determined by chain components. In particular, when every maximal chain of P is infinite, all additive biderivations are inner. Full article

Urban livability and sustainable development remain major global challenges, yet the interplay between urban planning layouts and actual human activities has not been sufficiently examined. This study investigates this relationship in Beijing by integrating multi-source spatiotemporal data, including point of interest (POI), Land Use Cover Change (LUCC), remote sensing data, and the railway network. Defining urban functional units as “street + railway network”, we analyze the spatial–temporal evolution within the 6th Ring Road over the past four decades and propose targeted strategies for the urban functional layout. The results reveal the following: (1) The evolution of Beijing’s urban functions can be divided into four stages (1980–1990, 1990–2005, 2005–2015, and 2015–2020), with continuous population growth (+142%) driving the over-concentration of functions in central districts. (2) Between 2010 and 2020, the POI densities of medical services (+133.6%) and transport services (+130.48%) increased most rapidly, subsequently stimulating the expansion of other urban functions. (3) High-density functional facilities and construction land (+179.10%) have expanded significantly within the 6th Ring Road, while green space (cropland, forestland and grassland) has decreased by 86.97%, resulting in a severe imbalance among land use types. To address these issues, we recommend the following: redistributing high-intensity functions to sub-centers such as Tongzhou and Xiongan New Area to alleviate population pressure, expanding high-capacity rail transit to reinforce 30–50 km commuting links between the core and periphery, and establishing ecological corridors to connect green wedges, thereby enhancing carbon sequestration and environmental quality. This integrated framework offers transferable insights for other megacities, providing guidance for sustainable functional planning that aligns human activity patterns with urban spatial structures. Full article

Let $\tau$ and $\sigma :X⟶X$ be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and $(\tau ,\sigma )$-L-derivations by introducing and analyzing a novel additive mapping $\Pi :X\to X$ called a generalized $(\tau ,\sigma )$-L-derivation associated with a $(\tau ,\sigma )$-L-derivation $\pi$. Later, we will examine the algebraic properties of a factor ring $X/L$ under the influence of certain algebraic expressions containing this generalized $(\tau ,\sigma )$-L-derivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated. Full article

Let $\mathfrak{R}$ be a commutative ring with identity $1\ne 0$. The weakly zero-divisor graph of $\mathfrak{R}$, denoted $W_{\Gamma }(\mathfrak{R})$, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of $\mathfrak{R}$, where two distinct vertices $\mathfrak{a}$ and $\mathfrak{b}$ are adjacent if and only if there exist $r\in \mathrm{ann}(\mathfrak{a})$ and $s\in \mathrm{ann}(\mathfrak{b})$ such that $rs=0$. In this paper, we study the signless Laplacian spectrum of $W_{\Gamma }({\mathbb{Z}}_n)$ for several composite forms of n, including $n=p^2{q}^2$, $n=p^{2}qr$, $n=p^m{q}^m$ and $n=p^{m}qr$, where $p, q, r$ are distinct primes and $m\ge 2$. By using generalized join decomposition and quotient matrix methods, we obtain explicit eigenvalue formulas for each case, along with structural bounds, spectral integrality conditions and Nordhaus–Gaddum-type inequalities. Illustrative examples with computed spectra are provided to validate the theoretical results, demonstrating the interplay between the algebraic structure of ${\mathbb{Z}}_n$ and the spectral properties of its weakly zero-divisor graph. Full article

Let us consider the finite commutative ring R, whose unity is $1\ne 0.$ Its weakly zero-divisor graph, represented as $W\Gamma \left(R\right)$, is a basic undirected graph with two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exist $r\in$ $\mathrm{ann}\left(c_1\right)$ and $s\in$ $\mathrm{ann}\left(c_2\right)$ that satisfy the condition $rs=0$. Let $D\left(G\right)$ be the distance matrix and $Tr\left(G\right)$ be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The $D_{\alpha }$ matrix of graph G is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$ for $\alpha \in [0,1]$. This article finds the $D_{\alpha }$ spectrum for the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ for various values of n and also shows that $W\Gamma \left({\mathbb{Z}}_n\right)$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes, is $D_{\alpha }$ integral. Full article

This paper presents the construction and analysis of a novel class of alternant codes over Gaussian integers, aimed at enhancing error correction capabilities in high-reliability communication systems. These codes are constructed using parity-check matrices derived from finite commutative local rings with unity, specifically ${\mathbb{Z}}_n\left[i\right],$ where $i^2=-1$. A detailed algebraic investigation of the polynomial $x^n-1$ over these rings is conducted to facilitate the systematic construction of such codes. The proposed alternant codes extend the principles of classical BCH and Goppa codes to complex integer domains, enabling richer algebraic structures and greater error-correction potential. We evaluate the performance of these codes in terms of error correction capability, and redundancy. Numerical results show that the proposed codes outperform classical short-length codes in scenarios requiring moderate block lengths, such as those applicable in certain segments of 5G and IoT networks. Unlike conventional codes, these constructions allow enhanced structural flexibility that can be tuned for various application-specific parameters. While the potential relevance to quantum-safe communication is acknowledged, it is not the primary focus of this study. This work demonstrates how extending classical coding techniques into non-traditional algebraic domains opens up new directions for designing robust and efficient communication codes. Full article

In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring S with unity. Consider a graph Γ(S) with vertex set V(Γ) = Z*(S), and two vertices in Γ(S) are adjacent if and only if their product is zero. This graph Γ(S) is known as zero-divisor graph of S. Zero-divisor graphs provide a powerful bridge between abstract algebra and graph theory. The zero-divisor graphs for finite commutative rings and their minimum fault-tolerant edge-resolving sets are studied in this article. Through analytical and constructive techniques, we highlight how the algebraic properties of the ring influence the edge metric structure of its associated graph. In addition to this, the existence of a connected graph G having a resolving set of cardinality of 2n + 2 from a star graph $K_{\mathrm{1,2}n}$, is studied. Full article

Let R be a finite commutative ring with nonzero identity 1. In this paper, domination parameters, the domination number and the total dominating number, of the unit graph $\mathrm{G}(R)$ or the closed unit graph $\overline{\mathrm{G}}(R)$ of R are investigated. To study the domination number of $\mathrm{G}(R)$, we prove that it suffices to consider the case when R is a direct product of fields. Furthermore, we discuss the domination number and total dominating number of the unit graph of the ring of integers modulo n. Full article

This paper extends the concept of the total graph $T_{\mathsf{\Gamma }}\left(R\right)$ associated with a commutative ring to the three-fold Cartesian product $R={\mathbb{Z}}_n\times {\mathbb{Z}}_m\times {\mathbb{Z}}_p$, where $n,m,p>1$. We present complete and self-contained proofs for a wide range of graph-theoretic properties of $T_{\mathsf{\Gamma }}\left(R\right)$, including connectivity, diameter, regularity conditions, clique and independence numbers, and exact criteria for Hamiltonicity and Eulericity. We also derive improved lower bounds for the genus and characterize the automorphism group in both general and symmetric cases. Each result is illustrated through concrete numerical examples for clarity. Beyond theoretical contributions, we discuss potential applications in cryptographic key-exchange systems, fault-tolerant network architectures, and algebraic code design. This work generalizes and deepens prior studies on two-factor total graphs, and establishes a foundational framework for future exploration of higher-dimensional total graphs over finite commutative rings. Full article

In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded weakly 2-absorbing ideals). Let G be a group and let R be a non-commutative G-graded ring with nonzero unity. Let P be a proper graded ideal of R. We then say that P is a graded 1-absorbing prime ideal (a graded weakly 1-absorbing prime ideal) of R if, for each nonunit homogeneous element $r,s,t\in R$ with $rRsRt\subseteq P$ ($\left\{0\right\}\ne rRsRt\subseteq P$), either $rs\in P$ or $t\in P$. We present a number of properties and characterizations of these graded ideals. Full article

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order $p^2$, where p is a prime satisfying $p\equiv 1mod4$ or $p\equiv 3mod4$. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods. Full article

Let $\Lambda$ denote a commutative ring with unity and $\mathfrak{D}\left(\Lambda \right)$ denote a collection of all annihilating ideals from $\Lambda$. An annihilator intersection graph of $\Lambda$ is represented by the notation $\mathcal{AIG}\left(\Lambda \right)$. This graph is not directed in nature, where the vertex set is represented by $\mathfrak{D}{\left(\Lambda \right)}^{*}$. There is a connection in the form of an edge between two distinct vertices $\varsigma$ and $\varrho$ in $\mathcal{AIG}\left(\Lambda \right)$ iff $Ann\left(\varsigma \varrho \right)\ne Ann\left(\varsigma \right)\cap Ann\left(\varrho \right)$. In this work, we begin by categorizing commutative rings $\Lambda$, which are finite in structure, so that $\mathcal{AIG}\left(\Lambda \right)$ forms a star graph/2-outerplanar graph, and we identify the inner vertex number of $\mathcal{AIG}\left(\Lambda \right)$. In addition, a classification of the finite rings where the genus of $\mathcal{AIG}\left(\Lambda \right)$ is 2, meaning $\mathcal{AIG}\left(\Lambda \right)$ is a double-toroidal graph, is also investigated. Further, we determine $\Lambda$, having a crosscap 1 of $\mathcal{AIG}\left(\Lambda \right),$ indicating that $\mathcal{AIG}\left(\Lambda \right)$ is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two. Full article

In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to have a reducible solution. Finally, we present an algorithm, along with two concrete examples to validate the main conclusions. Full article

This study investigates finite commutative local non-chain rings characterized by the well-established invariants $p, n, m, l$, and k, where p denotes a prime number. We specifically focus on Frobenius local rings with length $l=5$ and an index of nilpotency $t=4$. The significance of Frobenius rings in coding theory arises when specific results of linear codes are applicable to both finite fields and finite Frobenius rings. In light of this, we provide a comprehensive classification and enumeration of Frobenius local rings of order $p^{5m}$ with $t=4$, highlighting their distinctive properties in relation to varying values of $n.$ This research advances our understanding of the structural features of Frobenius rings and their applications in coding theory. Full article