The Co-annihilating-ideal Graphs of Commutative Rings (original) (raw)

The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

Communications in Algebra, 2014

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted R , is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted I R. We discuss when I R is bipartite. We also give some results on the subgraphs and the parameters of I R .

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 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 graph of a commutative ring

Discrete Mathematics, Algorithms and Applications, 2018

Let R be a commutative ring with identity and U R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by CA R , is a graph with vertex set U R and two vertices a and b are adjacent whenever Ann(a) ∩ Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

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 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).

The annihilating-ideal graph of commutative rings II

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.

The annihilator-inclusion Ideal graph of a commutative ring

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 RRR and two distinct vertices III and JJJ 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.

On the Annihilator Graph of a Commutative Ring

Communications in Algebra, 2013

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 R is an edge (path) of AG R. In this article, we study the graph AG R. For a commutative ring R, we show that AG R is connected with diameter at most two and with girth at most four provided that AG R has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG R is identical to the zero-divisor graph R if and only if R has exactly two minimal prime ideals.

Some Properties of Annihilator Graph of a Commutative Ring

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.