##### THE ANNIHILATOR GRAPH FOR MODULED OVER COMMUTATIVE RINGS

Katayoun Nozari; Sh. Payrovi

Volume 9, Issue 1 , September 2021, Pages 1-12
##### Abstract
‎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 ...  Read More

##### ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES

Y. Irani; K. Bahmanpour; Gh. Ghasemi

Volume 9, Issue 1 , September 2021, Pages 13-20
##### Abstract
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 ...  Read More

##### SOME PROPERTIES ON DERIVATIONS OF LATTICES

Mayuka F Kawaguchi; Michiro KONDO

Volume 9, Issue 1 , September 2021, Pages 21-33
##### Abstract
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 $x\in L$ (ii) a monotone derivation $d$ is characterized by $d(x) = x\wedge d(1)$ and (iii) ...  Read More

##### DEFICIENCY ZERO GROUPS IN WHICH PRIME POWER OF GENERATORS ARE CENTRAL

Volume 9, Issue 1 , September 2021, Pages 35-43
##### Abstract
The infinite family of groups defined by the presentation $G_p=\langle x, y|x^p=y^p,\; xyx^my^n=1\rangle$, in which $p$ is a prime in $\{2,3,5\}$ and $m,n\in\mathbb{N}_0$, will be considered and finite and infinite groups in the family will be determined. For the primes $p=2,3$ the group $G_p$ is finite ...  Read More

##### C#-IDEALS OF LIE ALGEBRAS

L. Goudarzi

Volume 9, Issue 1 , September 2021, Pages 45-51
##### Abstract
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 $H\cap K$ 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 ...  Read More

##### GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED

M. Alishahi; E. Rezaei-Sani

Volume 9, Issue 1 , September 2021, Pages 53-60
##### Abstract
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 ...  Read More

##### FUZZY MEDIAL FILTERS OF PSEUDO BE-ALGEBRAS

A. Rezaei

Volume 9, Issue 1 , September 2021, Pages 61-82
##### Abstract
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 ...  Read More

##### m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS

Volume 9, Issue 1 , September 2021, Pages 83-106
##### Abstract
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 ...  Read More

##### SOME RESULTS ON ϕ -(k,n)-CLOSED SUBMODULES

M. H. Moslemi Koupaei

Volume 9, Issue 1 , September 2021, Pages 107-118
##### Abstract
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 $\phi :S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function. A proper submodule $N$ of $M$ is called $\phi$ -semi-$n$-absorbing if $r^{n} m\in N\setminus \phi(N)$ ...  Read More

##### ALGORITHMIC ASPECTS OF ROMAN GRAPHS

A. Poureidi

Volume 9, Issue 1 , September 2021, Pages 119-135
##### Abstract
Let $G=(V, E)$ be a graph. A set $S \subseteq V$ is called a dominating set of $G$ if for every $v\in V-S$ there is at least one vertex $u \in N(v)$ such that $u\in S$. The domination number of $G$, denoted by $\gamma(G)$, is equal to the minimum cardinality of a dominating set in $G$. A Roman dominating ...  Read More