Ideal Graphs

Discrete Mathematics, Algorithms and Applications, 2016

Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

Journal of the Korean Mathematical Society, 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T (Γ I (R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ S(I). The total graph of a commutative ring, that denoted by T (Γ(R)), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ∈ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, T (Γ I (R)) = T (Γ(R)); this is an important result on the definition.

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

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) .

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).

Journal of the Korean Mathematical Society, 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zerodivisor graphs of semigroups. The first definition gives a directed graph Γ(S), and the other definition yields an undirected graph Γ(S). It is shown that Γ(S) is not necessarily connected, but Γ(S) is always connected and diam(Γ(S)) ≤ 3. For a ring R define a directed graph APOG(R) to be equal to Γ(IPO(R)), where IPO(R) is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph APOG(R) to be equal to Γ(IPO(R)). We show that R is an Artinian (resp., Noetherian) ring if and only if APOG(R) has DCC (resp., ACC) on some special subset of its vertices. Also, It is shown that APOG(R) is a complete graph if and only if either (D(R)) 2 = 0, R is a direct product of two division rings, or R is a local ring with maximal ideal m such that IPO(R) = {0, m, m 2 , R}. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings M n×n (R) where n ≥ 2.

Armenian journal of mathematics, 2023

In this paper, we relate some properties of noncomaximal graph of ideals of a commutative ring with identity with the properties of the ring.

2011

Let R be a commutative ring with A(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilatingideal graph of R, denoted by AG(R). It is the (undirected) graph with vertices A(R) * := A(R) \ {(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of AG(R). For instance, it is shown that if R is not a domain, then AG(R) has ACC (resp., DCC) on vertices if and only if R is Noetherian (resp., Artinian). Moreover, the set of vertices of AG(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, AG(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of AG(R). It is shown that AG(R) is a connected graph and diam(AG)(R) ≤ 3 and if AG(R) contains a cycle, then g(AG(R)) ≤ 4. Also, rings R for which the graph AG(R) is complete or star, are characterized, as well as rings R for which every vertex of AG(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

2012

In this paper, we determine the diameters of graphs Γ ′ 2 (R) and C (R) for a ring R with infinitely many maximal ideals. We also use graph blow-up to give a complete classification of rings R whose graphs C (R) are non-empty planar graphs.

Mathematics

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.