On the comaximal ideal graph of a commutative 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).
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.
A Note on Co-Maximal Ideal Graph of Commutative Rings
Let R be a commutative ring with unity. The co-maximal ideal graph of R, denoted by Γ(R), is a graph whose vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If Γ(R) is an infinite star graph, can R be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question. *
The Total Graph of a Commutative Ring with Respect to Proper Ideals
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.
A new graph associated to a commutative ring
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.
Non-comaximal graph of ideals of a ring
Proceedings - Mathematical Sciences, 2019
Let R be a ring. The non-comaximal graph of R, denoted by NC(R) is an undirected graph whose vertex set is the collection of all non-trivial (left) ideals of R and any two distinct vertices I and J are adjacent if and only if I + J = R. The concepts of connectedness, independent set, clique and traversability of NC(R) are discussed.
On the annihilator-ideal graph of commutative rings
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).
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
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.
SOME RESULTS ON THE JACOBSON GRAPH OF A COMMUTATIVE RING
Rend. Circ. Mat. Palermo, II. Ser, 2018
Let R be a commutative ring with non-zero identity and J(R) be Jacobson ideal of R. The Jacobson graph of R is the graph whose vertices are R\J(R), and two different vertices x and y are adjacent if 1−xy / ∈ U (R), where U (R) is the set of units of R. We investigate diameter of J R and seek relation between it and diameter of Jacobson graphs under extension to polynomial and power series rings. Also, vertex and edge connectivity of finite Jacobson graphs are obtained. Finally, we show that all finite Jacobson graphs have a matching that misses at most one vertex and offer one 1-factor decomposition of a regular subgraph.
The annihilator ideal graph of a commutative ring
2016
Let be a commutative ring with nonzero identity and be a proper ideal of . The annihilator graph of with respect to , which is denoted by , is the undirected graph with vertex-set for some and two distinct vertices and are adjacent if and only if , where . In this paper, we study some basic properties of , and we characterise when is planar, outerplanar or a ring graph. Also, we study the graph , where is the ring of integers modulo .
The Co-annihilating-ideal Graphs of Commutative Rings
Canadian Mathematical Bulletin, 2016
LetRbe a commutative ring with identity. The co-annihilating-ideal graph ofR, denoted byAR, is a graph whose vertex set is the set of all non-zero proper ideals ofRand two distinct verticesIandJare adjacent whenever Ann(I) ∩ Ann(J) = {0}. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.
The annihilating-ideal graph of commutative rings I
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.
The Annihilating-Ideal Graph of a Ring
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.
On the Structure of Comaximal Graphs of Commutative Rings with Identity
Bulletin of the Australian Mathematical Society, 2010
In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.
On the Regular Digraph of Ideals of Commutative Rings
Bulletin of the Australian Mathematical Society, 2013
Let RRR be a commutative ring. The regular digraph of ideals of RRR, denoted by Gamma(R)\Gamma (R)Gamma(R), is a digraph whose vertex set is the set of all nontrivial ideals of RRR and, for every two distinct vertices III and JJJ, there is an arc from III to JJJ whenever III contains a nonzero divisor on JJJ. In this paper, we study the connectedness of Gamma(R)\Gamma (R)Gamma(R). We also completely characterise the diameter of this graph and determine the number of edges in Gamma(R)\Gamma (R)Gamma(R), whenever RRR is a finite direct product of fields. Among other things, we prove that RRR has a finite number of ideals if and only if mathrmNGamma(R)(I)\mathrm {N}_{\Gamma (R)}(I)mathrmNGamma(R)(I) is finite, for all vertices III in Gamma(R)\Gamma (R)Gamma(R), where mathrmNGamma(R)(I)\mathrm {N}_{\Gamma (R)}(I)mathrmNGamma(R)(I) is the set of all adjacent vertices to III in Gamma(R)\Gamma (R)Gamma(R).
2018
The rings considered in this article are nonzero commutative with identity which are not fields. Let \(R\) be a ring. We denote the collection of all proper ideals of \(R\) by \(\mathbb{I}(R)\) and the collection \(\mathbb{I}(R)\setminus \{(0)\}\) by \(\mathbb{I}(R)^{*}\). Recall that the intersection graph of ideals of \(R\), denoted by \(G(R)\), is an undirected graph whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent if and only if \(I\cap J\neq (0)\). In this article, we consider a subgraph of \(G(R)\), denoted by \(H(R)\), whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent in \(H(R)\) if and only if \(IJ\neq (0)\). The purpose of this article is to characterize rings \(R\) with at least two maximal ideals such that \(H(R)\) is planar.
The principal ideal subgraph of the annihilating-ideal graph of commutative rings
2016
Let R be a commutative ring with identity and A(R) be the set of ideals of R with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of R, denoted by AGP (R). It is a (undirected) graph with vertices AP (R) = A(R) ∩ P(R) \ {(0)}, where P(R) is the set of proper principal ideals of R and two distinct vertices I and J are adjacent if and only if IJ = (0). Then, we study some basic properties of AGP (R). For instance, we characterize rings for which AGP (R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of AGP (R). Finally, we compare the principal ideal subgraph AGP (R) and spectrum subgraph AGs(R).