arXiv subjects
Publications and source records attributed to Ayman Badawi.
Let $H$ be a finite abelian (commutative) group of order $n \geq 2$, and $m >1$ be an integer. We define the $m$-graph of $H$, denoted by $m-G(H)$, as a simple undirected graph with vertex set $H$, and two distinct vertices, $a, b \in H$, are connected by an edge if and only if $a^m = b$ or $b^m = a$. Several results regarding the properties of the $m$-$G(H)$ have been established.
Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The $n$-total graph of $(R, D)$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n \in D$. In this paper, we study some graph properties and theoretical ring structure.
Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n$ in $Z(R)$. Note that if $n =1$, then the $1$-total graph of $R$ is the total graph of $R$ in the sense of Anderson-Badawi's paper on the total graph of a commutative ring. In this paper, we study some graph properties and theoretical ring structure.
Let $R$ be a commutative ring with $1 \neq 0$. A proper ideal $I$ of $R$ is a {\it square-difference factor absorbing ideal} (sdf-absorbing ideal) of $R$ if whenever $a^2 - b^2 \in I$ for $0 \neq a, b \in R$, then $a + b \in I$ or $a - b \in I$. In this paper, we introduce and investigate sdf-absorbing ideals.
For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or \emph{comparability graph} of $A$. Further, we say that a graph $G$ is a partial order graph if there exists a partially ordered set $A$ such that $G = G_A$. For a class $\mathcal{C}$ of simple, undirected graphs and $n$, $m \ge 1$, we define the Ramsey number $\mathcal{R}_{\mathcal{C}}(m,n)$ with respect to $\mathcal{C}$ to be the minimal number of vertices $r$ such that every induced subgraph of an arbitrary partial order graph consisting of $r$ vertices contains either a complete $n$-clique $K_n$ or an independent set consisting of $m$ vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.
Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.
Let R be a commutative ring with identity. In this paper, we introduce the concept of (m, n)-closed ideals of R and (m, n)-von Neumann regular rings
Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly semiprimary ideal} of $R$ if whenever $a,b\in R$ and $0 \not = ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let $I(R)$ be the set of all ideals of $R$ and let $δ: I(R) \rightarrow I(R)$ be a function. Then $δ$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J \subseteq I$, then $L \subseteq δ(L)$ and $δ(J) \subseteq δ(I)$. Let $δ$ be an expansion function of ideals of $R$. Then a proper ideal $I$ of $R$ (i.e., $I \not = R$) is called a ({\it $δ$-semiprimary}) {\it weakly $δ$-semiprimary} ideal of $R$ if ($ab \in I$) $0 \not = ab \in I$ implies $a \in δ(I)$ or $b \in δ(I)$. For example, let $δ: I(R) \rightarrow I(R)$ such that $δ(I) = \sqrt{I}$. Then $δ$ is an expansion function of ideals of $R$ and hence a proper ideal $I$ of $R$ is a ($δ$-semiprimary) weakly $δ$-semiprimary ideal of $R$ if and only if $I$ is a (semiprimary) weakly semiprimary ideal of $R$. A number of results concerning weakly $δ$-semiprimary ideals and examples of weakly $δ$-semiprimary ideals are given.
In this paper, we study the unit dot graph of product of commutative rings
In this paper, we introduce the concept of n-semiprimary ideals, n-powerful ideals, and n-powerful semiprimary ideals of commutative rings. We study these concepts and relate them to several generalizations of pseudo-valuation domains.
Let R be a commutative ring with $1\neq0$. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing ideal. A proper ideal $I$ of $R$ is called a weakly 1-absorbing primary ideal if whenever nonunit elements $a,b,c\in R$ and $0\neq abc\in I,$ then $ab\in I$ or $c\in\sqrt{I}$. A number of results concerning weakly 1-absorbing primary ideals and examples of weakly 1-absorbing primary ideals are given. Furthermore, we give the correct version of a result on 1-absorbing ideals of commutative rings.
Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets arise in so many natural examples. For example, let D be the set of all odd integers, then (D, +) is a 3-closed subset of (Z, +) that is not a 2-closed subset of (Z, +). If K = {1, 4, 7, 10, ...}, then (K, +) is a 4-closed subset of (Z, +) that is not an n-closed subset of (Z, +) for n = 2, 3. In this paper, we show that if (H, *) is a subgroup of a group (G, *) such that [H: G] = n < infty, then H is a normal subgroup of G if and only if every left coset of $H$ is an (n+1)-closed subset of G.