EULERIAN GRAPH OF SOME SPECIAL IDEALIZATION RINGS

Journal of Pure and Applied Algebra, 2006

We consider zero-divisor graphs of idealizations of commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the zero-divisor graph of a ring when extending to idealizations of the ring.

Communications in Algebra, 2006

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ 0 (R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ I (R) ∼ = Γ J (S) and Γ(R/I) ∼ = Γ(S/J). We also discuss when Γ I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ I (R).

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of R, denoted by C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I ̸⊆ J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C(R) if and only if I1 + I2 = R. In Section 2 of this article, we classify rings R such that C (R) is planar. In Section 3 of this article, we classify rings R such that C (R) is a split graph. In Section 4 of this article, we classify rings R such that C(R) is complemented and moreover, we determine the S-vertices of C (R).

TURKISH JOURNAL OF MATHEMATICS, 2016

Let R be a commutative ring with identity. We use Γ(R) to denote the comaximal ideal graph. The vertices of Γ(R) are proper ideals of R that are not contained in the Jacobson radical of R , and two vertices I and J are adjacent if and only if I + J = R. In this paper we show some properties of this graph together with the planarity and perfection of Γ(R) .

Journal of Algebra and Related Topics, 2016

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this paper we extend this fact for certain noncommutative rings, for example, reduced rings, right (left) self-injective rings and one-sided Artinian rings. The necessary and sufficient conditions for two reduced right Goldie rings to have isomorphic zero-divisor graphs is given. Also, we extend some known results about the zero-divisor graphs from the commutative to noncommutative setting: in particular, complemented and uniquely complemented graphs.

Algebra Colloquium, 2014

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by − − → Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian

Journal of the Korean Mathematical Society, 2014

Let R be a commutative ring with identity and M an Rmodule. In this paper, we associate a graph to M , say Γ(M), such that when M = R, Γ(M) is exactly the classic zero-divisor graph. Many wellknown results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in the present article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3. We also show that for a reduced module M with Z(M) * = M \ {0}, gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, x, y ∈ M \ {0} are adjacent if and only if xR yR = (0). Among other things, it is also observed that Γ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, and Z(M) * = M \ {0}, if and only if ann(M) is prime and Z(M) * = M \ {0}.

Communications in Algebra, 2008

Let R be a commutative ring with identity, Z(R) its set of zerodivisors, and N il(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R) \ {0}, with distinct vertices x and y adjacent if and only if xy = 0. In this paper, we study Γ(R) for rings R with nonzero zerodivisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} = N il(R) ⊆ zR for all z ∈ Z(R) \ N il(R).

Ricerche di Matematica, 2016

Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if ann(x) ∪ ann(y) = ann(x y), where for t ∈ R, we set ann(t) := {r ∈ R | rt = 0}. In this paper, we define the annihiator-ideal graph of R, which is denoted by A I (R), as an undirected graph with vertex set A * (R), and two distinct vertices I and J are adjacent if and only if ann(I) ∪ ann(J) = ann(I J). We study some basic properties of A I (R) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and A I (R) are coincide. Moreover, we examin the planarity of the graph A I (R).

Let R(+)N be the idealization of the ring R by the R-module N. In this paper, we investigate when Γ(R(+)N) is a Planar graph where R is an integral domain and we investigate when Γ(Z n (+)Z m) is a Planar graph.