The regular graph of a commutative ring

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2 ⁿ , where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.     

The M-principal graph of a commutative ring

Let \(R\) be a commutative ring and \(M\) be an \(R\) -module. In this paper, we introduce the \(M\) -principal graph of \(R\) , denoted by \(M-PG(R)\) . It is the graph whose vertex set is \(R\backslash \{0\}\) , and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xM=yM\) . In the special case that \(M=R, M-PG(R)\) is denoted by \(PG(R)\) . The basic properties and possible structures of these two graphs are studied. Also, some relations between \(PG(R)\) and \(M-PG(R)\) are established.     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

The trace graph of the matrix ring over a finite commutative ring

Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\), which we call the trace graph of the matrix ring \({M_n(R)}\), such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.     

Some results on abstract commutative ideal theory

Conditions are given for a multiplicative lattice to be a finite Boolean algebra. Multiplicative lattices in which semiprimary elements are primary or in which prime elements are weak meet principal are studied. The lattice of filters of a bounded commutative semilattice are investigated. Finally, we study compactly packed lattices.     

On annihilator ideals of a polynomial ring over a noncommutative ring

Let R be a ring and let R[x] denote the polynomial ring over R. We study relations between the set of annihilators in R and the set of annihilators in R[x].     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.     

Some results on the incidence coloring number of a graph

This paper proves that if G is a cubic graph which has a Hamiltonian path or G is a bridgeless cubic graph of large girth, then its incidence coloring number is at most 5. By relating the incidence coloring number of a graph G to the chromatic number of G², we present simple proofs of some known results, and characterize regular graphs G whose incidence coloring number equals Δ(G)+1.     

The zero-divisor graph of a commutative semigroup

An undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0. The vertices of the graph are labeled by the nonzero zero-divisors of S , and two vertices x,y are connected by an edge in case xy = 0 in S . The properties and possible structures of the graph Γ (S) are studied.     

Some fixed point results on a metric space with a graph

Combining some branches is a typical activity in different fields of science, especially in mathematics. Naturally, it is notable in fixed point theory. Over the past few decades, there have been a lot of activity in fixed point theory and another branches in mathematics such differential equations, geometry and algebraic topology. In 2006, Espinola and Kirk made a useful contribution on combining fixed point theory and graph theory. Recently, Reich and Zaslavski studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, by using main idea of their work and the idea of combining fixed point theory and graph theory, we present some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs.