Journal of Algebraic Systems
Volume:11 Issue: 1, Summer-Autumn 2023

For a commutative ring $R$ with identity $1\neq 0$, let $Z^{*}(R)=Z(R)\setminus \lbrace 0\rbrace$ be the set of non-zero zero-divisors of $R$, where $Z(R)$ is the set of all zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*}(R)=Z(R)\setminus \{0\}$ and two vertices of $ Z^*(R)$ are adjacent if and only if their product is $ 0 $. In this article, we find the structure of the zero-divisor graphs $ \Gamma(\mathbb{Z}_{n}) $, for $n=p^{N_1}q^{N_2}r$, where $2<p<q<r$ are primes and $N_1$ and $N_2$ are positive integers.

In this paper, we introduce the concept of  module Lie  derivations on Banach algebras and study  module Lie  derivations on unital triangular Banach algebras $ \mathcal{T}=\begin{bmatrix}A & M\\ &B\end{bmatrix}$ to its dual. Indeed, we prove that every module (linear) Lie derivation\linebreak $ \delta: \mathcal{T} \to \mathcal{T}^{\ast}$  can be decomposed as $ \delta = d + \tau $, where $ d: \mathcal{T} \to \mathcal{T}^{\ast} $ is a module (linear) derivation and $ \tau: \mathcal{T} \to Z_{\mathcal{T}}(\mathcal{T}^{\ast}) $  is a module (linear) map vanishing at commutators if and only if this happens for the corner algebras $A$ and $B$.

‎Let $R$ be a commutative Noetherian ring with non-zero identity and $\mathcal{F}$ a filtration of $\operatorname{Spec}(R)$‎. ‎We show that the Cousin functor with respect to $\mathcal{F}$‎, ‎$C_R(\mathcal{F},-):\mathcal{C}_{\mathcal{F}}(R)\longrightarrow\operatorname{Comp}(R)$‎, ‎where $\mathcal{C}_{\mathcal{F}}(R)$ is the category of $R$-modules which are admitted by $\mathcal{F}$ and $\operatorname{Comp}(R)$ is the category of complexes of $R$-modules‎, ‎commutes with the formation of direct limits and is right exact‎. ‎We observe that an $R$-module $X$ is balanced big Cohen-Macaulay if $(R,\mathfrak{m})$ is a local ring‎, ‎$\mathfrak{m}X\neq X$‎, ‎and every finitely generated submodule of $X$ is a big Cohen-Macaulay $R$-module with respect to some system of parameters for $R$‎.

‎Suppose that $G$ is a finite group. ‎The acentralizer $C_G(\alpha)$ of an automorphism $\alpha$ of $G$‎,‎is defined as the subgroup of fixed points of $\alpha$‎, ‎that is $C_G(\alpha)= \{g \in G \mid \alpha(g)=g\}$‎.‎In this paper we determine the acentralizers of groups of order $p^3$‎, ‎where $p$ is a prime number.

The concepts of intrinsic ideals and inlets are introduced in a distributive lattice. Intrinsic ideals are also characterized with the help of inlets. Certain equivalent conditions are given for an ideal of a distributive lattice to become intrinsic. Some equivalent conditions are derived for the quotient lattice, with respect to a congruence, to become a Boolean algebra. Some topological properties of the prime spectrum of intrinsic ideals of distributive lattice are derived.

Let $G=(V(G),E(G))$ be a simple graph. A set $D\subseteq V(G)$ is a strong dominating set of $G$, if for every vertex $x\in V(G)\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $\gamma_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. In this paper, we calculate $\gamma_{st}(G)$ for specific graphs and study the number of strong dominating sets of some graphs.

In this article, we characterize $\{g, h\}$-derivation on the upper triangular matrix algebra $\mathcal{T}_n(C)$ and prove that every Jordan $\{g, h\}$-derivation over $\mathcal{T}_n(C)$ is a $\{g, h\}$-derivation under a certain condition, where $C$ is a $2$-torsion free commutative ring with unity $1\neq 0$. Also, we study $\{g, h\}$-derivation and Jordan $\{g, h\}$-derivation over full matrix algebra $\mathcal{M}_n(C)$.

Let $I$ be an ideal of a commutative Noetherian ring $R$ and $M$ be a non-zero Artinian $R$-module with support contained in $V(I)$. In this paper it is shown that $M$ is $I$-cofinite if and only if $Rad(I\widehat{R}^J+Ann_{\widehat{R}^J}M)=J\widehat{R}^J$, where $J:=\cap_{m\in Supp M}m$ and $\widehat{R}^J$ denotes the $J$-adic comletion of $R$.

The aim of this study is to introduce (anti) fuzzy ideals of a Sheffer stroke BCK-algebra. After describing an anti fuzzy subalgebra and an anti fuzzy (sub-implicative) ideal of a Sheffer stroke BCK-algebra, the relationships of these structures are demonstrated. Also, a t-level cut and a complement of a fuzzy subset are defined and some properties are investigated. An implicative Sheffer stroke BCK-algebra is defined and it is proved that a fuzzy subset of an implicative Sheffer stroke BCK-algebra is an anti fuzzy ideal if and only if it is an anti fuzzy sub-implicative ideal of this algebraic structure. A fuzzy congruence and a fuzzy quotient set of a Sheffer stroke BCK-algebra are studied in details and it is shown that there is a bijection between the set of fuzzy ideals and the set of fuzzy congruences on this algebraic structure. Finally, Cartesian product of fuzzy subsets of a Sheffer stroke BCK-algebra is determined and it is expressed that the Cartesian product of two anti fuzzy ideals of this algebraic structure is anti fuzzy ideal.

The paper studies the existence of linear codes that locate solid burst errors, which may be confined to one sub-block or spread over two adjacent sub-blocks. An example of such a code is also given. Comparisons on the number of parity check digits required for such linear codes with solid burst detecting and correcting codes are also provided.

In this paper, we partially generalize a result of Isbell from the class of commu- tative semigroups to some generalized class of commutative semigroups by showing that dominion of such semigroups belongs to the same class by using Isbell’s zigzag theorem. we found some permutative semigroups for which dominion satisfies the identity of subsemigroup of a semigroup S.

Let a be an ideal of local ring (R;m) and M a nitely generated R-module. Inthis paper, we prove some results concerning niteness and minimaxness of formal local cohomologymodules. In particular, we investigate some properties of top formal local cohomologyFdimM=aMa (M) and we determine CosR(FdimM=aMa (M)), AnnR(FdimM=aMa (M)) andAttR(FdimM=aMa (M)).