JOURNAL OF ALGEBRAIC SYSTEMS
Year:2025 | Volume:13 | Issue:2|Papers Count:12

‎Let $G=(V‎, ‎E)$ be a simple graph‎. ‎A set $C$ of vertices of $G$ is an identifying code of $G$ if for every two vertices $x$ and $y$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different‎. ‎Given a graph $G,$ the smallest size of an identifying code of $G$ is called the identifying code number of $G$ and denoted by $\gamma^{ID}(G).$ In this paper‎, ‎we prove that the identifying code number of the subdivision of a graph $G$ of order $n$ is at most $n$‎. ‎Also‎, ‎we prove that the identifying code number of the subdivision of graphs $K_n$, $K_{r,s}$ and $C_P(s)$ are $n‎$,‎ ‎‎‎$‎‎r+s$ and $2s$, respectively‎. ‎Finally‎, ‎we conjecture that for every graph $G$ of order $n$ the identifying code number of the subdivision of $G$ is $n$‎.

The notion of hemicomplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemicomplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemicomplemented lattice to become a D-Stone lattice. Hemicom-complemented lattices and D-Stone lattices are characterized in topological terms.

The notion of w−filters is introduced in an Almost Distributive Lattice(ADL) and properties are investigated. A necessary and sufficient condition is derived for a maximal filter of an ADL to become a w−filter which leads to a characterization of a quasi-complemented ADL. Also, w−filters of an ADL are characterized in terms of minimal prime D−filters.

An ideal $I$ of a ring $R$ is called a right strongly Baer ideal if $r(I)=r(e)$, where $e$ is an idempotent, and there are right semicentral idempotents $e_{i}$ ($1\leq i\leq n$) with $ReR=Re_{1}R\cap Re_{2}R\cap...\cap Re_{n}R$ and each ideal $Re_{i}R$ is maximal or equals $R$. In this paper, we provide a topological characterization of this class of ideals in semiprime (resp., semiprimitive) rings. By using these results, we prove that every ideal of a ring $R$ is a right strongly Baer ideal \textit{if and only if} $R$ is a semisimple ring. Next, we give a characterization of right strongly Baer-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, and semiprime rings. For a semiprimitive commutative ring $R$, it is shown that $\Soc(R)$ is a right strongly Baer ideal \textit{if and only if} the set of isolated points of $\Max(R)$ is dense in it \textit{if and only if} $\Soc_{m}(R)$ is a right strongly Baer ideal. Finally, we characterize strongly Baer ideals in $C(X)$ (resp., $C(X)_{F}$).

Let $G = (V, E)$ be a simple graph of order $n$. A total dominating set of $G$ is a subset $D$ of $V$ such that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to the minimum cardinality of a total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}_t$-equivalent to $G$. A polynomial $\sum_{k=0}^n a_kx^k$ is called unimodal if the sequence of its coefficients is unimodal, that means there is some $k \in \{0, 1, \ldots , n\}$, such that $a_0 \leq \ldots \leq a_{k-1} \leq a_k\geq a_{k+1} \geq \ldots \geq a_n$. In this paper, we investigate $\mathcal{D}_t$-equivalence classes of some graphs. Also, we introduce some families of graphs whose total domination polynomials are unimodal. The $\mathcal{D}_t$-equivalence classes of graphs of order $\leq 6$ are presented in the appendix.

We introduce the concept of a weak idempotent nil-clean ring as a generalization of a weakly nil-clean ring. We give certain characterizations for weak idempotent nil-clean rings in terms of Jacobson radical and nil radical. Further, we obtain any weak idempotent nil-clean ring is a direct product nil clean rings in terms of Jacobson radicals .

In this research paper, we introduce and analyze the notion of fuzzy neutrosophic prime ideals (FNPIs) in a commutative BCK-algebra $\mathcal{K}$. It represents a further extension of prime ideals in the context of fuzzy neutrosophic sets. We provide an example that shows that not every fuzzy neutrosophic ideal of a commutative BCK-algebra $\mathcal{K}$ is a FNPI of $\mathcal{K}$. We also prove that a fuzzy neutrosophic set of $\mathcal{K}$ is a FNPI of $\mathcal{K}$ if, for all $a,b,c \in [0,1] $, the upper (a,b)-level cut and lower c-level cut are prime ideals of $\mathcal{K}$.

A ring $R$ is called a left $\mathcal{Z}$-symmetric ring if $ab \in \mathcal{Z}_l(R)$ implies $ba \in \mathcal{Z}_l(R)$, where $\mathcal{Z}_l(R)$ is the set of left zero-divisors. A right $\mathcal{Z}$-symmetric ring is defined similarly, and a $\mathcal{Z}$-symmetric ring is one that is both left and right $\mathcal{Z}$-symmetric. In this paper, we introduce the concept of $\mathcal{Z}$-symmetric modules as a generalization of $\mathcal{Z}$-symmetric ring. Additionally, we introduce the concept of an eversible module as an analogy to eversible rings and prove that every eversible module is also a $\mathcal{Z}$-symmetric module, like in the case of rings.

This article embodies a ring theoretic property which, preserves the reversibility of elements at non-zero tripotents. A ring R is defined as quasi tri reversible if any non-zero tripotent element ab of R implies ba is also a tripotent element in R for a, b ∈ R. We explore the quasi tri reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, deducing that the quasi tri reversibility is a proper generalization of reversible rings. It is proved that the polynomial rings are not quasi tri reversible rings. The relation of symmetric rings, IF P and Abelian rings with reversibility and quasi tri reversibility arestudied. It is also observed that the structure of weakly tri normal rings and quasi tri reversible rings are independent of each other.

A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.

Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m<\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) graphs, total nilpotent dot product graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in \mathcal{N(R)}, \mbox{for some }z\in \mathcal{R}^*\} $. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in \mathcal{N}(\mathscr{B})$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N}(\mathscr{B})$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.

The commuting graph of a finite group G is a graph whose vertex set is the set of non-central elements of G and two distinct vertices are adjacent if they commute. In this article, we compute genus of commuting graphs of certain classes of finite non-abelian groups and characterize those groups such that their commuting graphs have genus 4, 5 and 6.