Journal of Algebraic Systems
Volume:9 Issue: 1, Summer-Autumn 2021

‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎The‎ ‎annihilator graph of $M$‎, ‎denoted by $AG(M)$ is a simple undirected‎ ‎graph associated to $M$ whose the set of vertices is‎ ‎$Z_R(M) setminus {rm Ann}_R(M)$ and two distinct vertices $x$ and‎ ‎$y$ are adjacent if and only if ${rm Ann}_M(xy)neq {rm‎ ‎Ann}_M(x) cup {rm Ann}_M(y)$‎. ‎In this paper‎, ‎we study the‎ ‎diameter and the girth of $AG(M)$ and we characterize all modules‎ ‎whose annihilator graph is complete‎. ‎Furthermore‎, ‎we look for the‎ ‎relationship between the annihilator graph of $M$ and its zero-divisor‎ ‎graph‎.

Let $(R, mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $mathrm{pd}_R, A=dim R$. Furthermore, as an application of these results it is shown that$$pdd H^i_{{frak p}R_{frak p}}(M_{frak p}, N_{frak p})leq pd H^{i+dim R/{frak p}}_{frak m}(M,N)$$for each ${frak p}in mathrm{Spec} R$ and each integer $igeq 0$. This result answers affirmatively a question raised by the present authors in [13].

In this paper we consider some properties of derivations of lattices and show that (i) for a derivation $d$ of a lattice $L$ with the maximum element $1$, it is monotone if and only if $d(x) le d(1)$ for all $xin L$ (ii) a monotone derivation $d$ is characterized by $d(x) = xwedge d(1)$ and (iii) simple characterization theorems of modular lattices and of distributive lattices are given by derivations. We also show that, for a distributive lattice $L$ and a monotone derivation $d$ of it, the set ${rm Fix}_d(L)$ of all fixed points of $d$ is isomorphic to the lattice $L/ker (d)$. We provide a counter example to the result (Theorem 4) proved in [3].

Let $L$ be a finite dimensional Lie algebra. A subalgebra $H$ of $L$ is called a $c^{#}$-ideal of $L$, if there is an ideal $K$ of $L$ with $L=H+K$ and $Hcap K$ is a $CAP$-subalgebra of $L$. This is analogous to the concept of a $c^{#}$-normal subgroup of a finite group. Now, we consider the influence of this concept on the structure of finite dimentional Lie algebras.

A subset of the vertex set of a graph $G$ is called a zero forcing set if by considering them colored and, as far as possible, a colored vertex with exactly one non-colored neighbor forces its non-colored neighbor to get colored, then the whole vertices of $G$ become colored. The total forcing number of a graph $G$, denoted by $F_t(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a subgraph with no isolated vertex. The connected forcing number, denoted by $F_c(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a connected subgraph. In this paper, we first characterize the graphs with $F_t(G)=2$ and, as a corollary, we characterize the graphs with $F_c(G)=2$.

In this paper, the notion of fuzzy medial filters of a pseudo BE-algebra is defined, and some of the properties are investigated. We show that the set of all fuzzy medial filters of a pseudo BE-algebra is a complete lattice. Moreover, we state that in commutative pseudo BE-algebras fuzzy filters and fuzzy medial filters coincide. Finally, the notion of a fuzzy implicative filter is introduced and proved that every fuzzy implicative filter is a fuzzy medial filter, and we show that the converse is not valid in general.

In this article we consider the $m$-topology on linebreak $M(X,mathscr{A})$, the ring of all real measurable functions on a measurable space $(X, mathscr{A})$, and we denote it by $M_m(X,mathscr{A})$. We show that $M_m(X,mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,mathscr{A})‎cong‎ mathbb R^n$ as topological rings. Also, we show that $M_m(X,mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,mathscr{A})= {M}_{u}(X,mathscr{A})$, if and only if $ M_m(X,mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,mathscr{A})$ is a connected space, if and only if $M_m(X,mathscr{A})$ is a locally connected space, if and only if $M^*(X,mathscr{A})$ is a connected subset of $M_m(X,mathscr{A})$.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$ -module. Let $S(M)$ be the set of all submodules of $M$ and $phi :S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function. A proper submodule $N$ of $M$ is called $phi$ -semi-$n$-absorbing if $r^{n} min Nsetminus phi(N)$ where $rin R, min M$ and $nin {Bbb Z}^+$, then $r^{n} in (N:M)$ or $r^{n-1} min N$. Let $k$ and $n$ are positive integers where $k>n$. A proper submodule $N$ of $M$ is called $phi$ -$(k,n)$- closed submodule, if $ r^{k}min Nsetminus phi(N)$ where $rin R$, $min M$ and $kin {Bbb Z}^+$, then $r^{n}in (N:M)$ or $r^{n-1}min N$. In this work, firstly, we will study some general results when we use the definition $phi$ -$(k,n)$- closed submodule. Moreover, we prove main results of the $phi$ -$(k,n)$- closed submodule for various modules.

Let $G=(V, E)$ be a graph. A set $S subseteq V$ is called a dominating set of $G$ if for every $vin V-S$ there is at least one vertex $u in N(v)$ such that $uin S$. The domination number of $G$, denoted by $gamma(G)$, is equal to the minimum cardinality of a dominating set in $G$. A Roman dominating function (RDF) on $G$ is a function $f:Vlongrightarrow{0,1,2}$ such that every vertex $vin V$ with $f(v)=0$ is adjacent to at least one vertex $u$ with $f(u)=2$. The weight of $f$ is the sum $f(V)=sum_{vin V}f (v)$. The minimum weight of a RDF on $G$ is the Roman domination number of $G$, denoted by $gamma_R(G)$. A graph $G$ is a Roman Graph if $gamma_R(G)=2gamma(G)$. In this paper, we first study the complexity issue of the problem posed in [E.J. Cockayane, P.M. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, On Roman domination in graphs, textit{Discrete Math.} 278 (2004), 11--22], and show that the problem of deciding whether a given graph is a Roman graph is NP-hard even when restricted to chordal graphs. Then, we give linear algorithms that compute the domination number and the Roman domination number of a given unicyclic graph. Finally, using these algorithms we give a linear algorithm that decides whether a given unicyclic graph is a Roman graph.

In the present manuscript, we introduce a type of hyper K-algebra which is called left absorbing hyper K-algebra and investigate some of the related properties. We also show that set of all types of positive implicative and commutative hyper K-ideal form a distributive latttice and study their diagrams when positive implicative and commutative hyper K-ideal are a hyper K-ideal and the hyper K-algebra is left absorbing.

It is well-known that, using principal weak flatness property, some important monoids are characterized, such as regular monoids, left almost regular monoids, and so on. In this article, we recall a generalization of principal weak flatness called GP-flatness, and characterize monoids by this property of their right (Rees factor) acts. Also we investigate GP coherent monoids.