The annihilator ideal graph of a commutative ring

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.

Abstract: Let R be a commutative ring with unity. Let Z(R) be the set of all zero-divisors of R. For x  Z(R), let ann (x) {yR | yx  0} R . We define the annihilator graph of R, denoted by ANNG(R), as the undirected graph whose set of vertices is Z(R)* = Z(R)  {0}, and two distinct vertices x and y are adjacent if and only if ann (xy) ann (x) ann (y) R R R   . In this paper, we study the ring-theoretic properties of R and the graph-theoretic properties of ANNG(R). For a commutative ring R, we show that ANNG(R) is connected, the diameter of ANNG(R) is at most two and the girth of ANNG(R) is at most four provided that ANNG(R) has a cycle. For a reduced commutative ring R, we study some characteristics of the annihilator graph ANNG(R) related to minimal prime ideals of R. Moreover, for a reduced commutative ring R, we establish some equivalent conditions which describe when ANNG(R) is a complete graph or a complete bipartite graph or a star graph. Keywords: Annihilator graph, diameter, girth, zero-divisor graph. 2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C25, 05C38, 05C40.

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 zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}(S)$, and the other definition yields an undirected graph $\overline{{\Gamma}}(S)$. It is shown that $\Gamma(S)$ is not necessarily connected, but $\overline{{\Gamma}}(S)$ is always connected and ${\rm diam}(\overline{\Gamma}(S))\leq 3$. For a ring $R$ define a directed graph $\Bbb{APOG}(R)$ to be equal to $\Gamma(\Bbb{IPO}(R))$, where $\Bbb{IPO}(R)$ is a semigroup consisting of all products of two one-sided ideals of $R$, and define an undirected graph $\overline{\Bbb{APOG}}(R)$ to be equal to $\overline{\Gamma}(\Bbb{IPO}(R))$. We show that $R$ is an Artinian (resp., Noetherian) r...

Transactions on Combinatorics, 2017

‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.

2011

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [4]). Let R be a commutative ring with A(R) its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph AG(R) that its vertices are A(R) * = A(R)\ {(0)} in which for every distinct vertices I and J, I −−−J is an edge if and only if IJ = (0). First, we study the diameter of AG(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(AG(R)) ≤ 2 or R is reduced and χ(AG(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(AG(R)) = cl(AG(R)). Moreover, if χ(AG(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(AG(R)) = cl(AG(R)) = n. Finally, we show that for a Noetherian ring R, cl(AG(R)) is finite if and only if for every ideal I of R with I 2 = (0), I has finite number of R-submodules.

Czechoslovak Mathematical Journal, 2017

2021

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

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).

2021

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilati...

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.

2016

Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set of all non-zero annihilating ideals and twovertices $I$ and $J$ are adjacent whenever ${rm Ann}(I)+{rmAnn}(J)$ is an essential ideal. In this paper we initiate thestudy of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or one.

Discrete Mathematics, 2012

Suppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) * = A(R) \ {(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). In Behboodi and Rakeei [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R)) = 3. Here, we prove that for every (not necessarily reduced) ring R, ω(AG(R)) ≥ |Min(R)|, which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate commutative rings whose annihilating-ideal graphs are bipartite. It is proved that AG(R) is bipartite if and only if AG(R) is triangle-free.

2017

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R) * = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) = ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). The extended zero-divisor graph of R is the undirected (simple) graph EG(R) with the vertex set Z(R) * , and two distinct vertices x and y are adjacent if and only if either Rx ∩ ann R (y) = {0} or Ry ∩ ann R (x) = {0}. Hence it follows that the zero-divisor graph Γ(R) is a subgraph of EG(R). In this paper, we collect some properties (many are recent) of the two graphs AG(R) and EG(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) .

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).