Non-comaximal graph of ideals of a ring (original) (raw)

Some results on the comaximal ideal graph of a commutative ring

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).

On the comaximal ideal graph of a commutative ring

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 Annihilating-Ideal Graph of Commutative Rings I

Journal of Algebra and Its Applications, 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 annihilating-ideal 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 ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, 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...

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

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

The Nil-Graph of Ideals of a Commutative Ring

Bulletin of the Malaysian Mathematical Sciences Society, 2015

Let R be a commutative ring with identity and Nil(R) be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph AG N (R) whose vertex set is {I : (0) = I R and there exists a non-trivial ideal J such that IJ ⊆ Nil(R)} and two distinct vertices I and J are adjacent if and only if IJ ⊆ Nil(R). Here, some graph properties of AG N (R) are studied. For instance, some bounds for the diameter, girth and the radius of AG N (R) are given. In case that AG N (R) is a finite graph, it is proved that the center and median of AG N (R) coincide. Furthermore, we determine when the edge chromatic number of AG N (R) equals its maximum degree. Also, for every ring R, it is shown that both the clique number and vertex chromatic number of AG N (R) equal n + t, where n is the number of minimal prime ideals of R and t is the number of non-zero ideals of R which are contained in Nil(R).

Independent sets of some graphs associated to commutative rings

2013

Let G = (V, E) be a simple graph. A set S ⊆ V is independent set of G, if no two vertices of S are adjacent. The independence number α(G) is the size of a maximum independent set in the graph. Let R be a commutative ring with nonzero identity and I an ideal of R.

Comaximal graph of commutative rings

Journal of Algebra, 2008

Let R be a commutative ring with identity. Let Γ(R) be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper we consider a subgraph Γ 2 (R) of Γ(R) which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph Γ 2 (R) \ J(R). In addition, it is shown that for two finite semi-local rings R and S, if R is reduced, then Γ(R) ∼ = Γ(S) if and only if R ∼ = S.

Comaximal Ideal Graphs of Commutative Rings

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.