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Central sets in the annihilating-ideal graph of commutative rings
Abstract and Figures
Let R be a commutative ring with identity and A (R) be the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) and two distinct vertices I 1 and I2 are adjacent if and only if I1I 2 = (0). In this paper, we study some connections between the graph-theoretic properties of AG(R) and algebraic properties of commutative ring R.
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... The annihilating ideal graph of commutative rings has also been studied by several researchers in [23], [7], [20], [21] and [19]. In [23], the authors have studied some connections between the graph-theoretic properties of AG(R) and algebraic properties of commutative ring R. In [7] some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring are presented. ...
... The annihilating ideal graph of commutative rings has also been studied by several researchers in [23], [7], [20], [21] and [19]. In [23], the authors have studied some connections between the graph-theoretic properties of AG(R) and algebraic properties of commutative ring R. In [7] some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring are presented. Also, the authors have investigated commutative rings whose annihilating-ideal graphs are complete or bipartite. ...
The intent of this dissertation is to study the interplay of ring-theoretic properties with graph theoretic properties. In this dissertation, we discuss in detail the results proved in the following papers:
1.M. Behboodi and Z. Rakeei, The annihilating ideal graphs ofcommutative rings I, J. Algebra Appl.
2. S. Visweswaran and Hiren D. Patel, On the Clique number of thecomplement of the annihilating ideal graph of a commutative ring,Beitr Algebra Geom.
3. S. Visweswaran and Hiren D. Patel, Some results on the complementof the annihilating ideal graph of a commutative ring, J. Algebra Appl.
4. S. Visweswaran and Pravin Vadhel, On the Dominating sets of thecomplement of the annihilating ideal graph of a commutative ring,Gulf Journal of Mathematics.
... AG(R) is a graph whose vertices are ideals of R with nonzero annihilators and in which two vertices I and J are adjacent if and only if IJ = (0). Later, it was modified and further studied by many authors (see [1,2,3,20,22]). ...
Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple undirected graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(\tau_T)$ which was introduced in [H. Ansari-Toroghy and S. Habibi, Comm. Algebra, 42(2014), 3283-3296]. In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.
... ere are many papers on assigning graphs to rings, groups, and semigroups [1][2][3][4][5][6]. Several authors [7][8][9][10][11][12][13] studied different properties of these graphs including diameter, girth, domination, metric dimension, central sets, and planarity. ...
Let be a numerical semigroup and be an irreducible ideal of . The graph assigned to an ideal of is a graph with elements of as vertices, and any two vertices are adjacent if and only if . In this work, we give a complete characterization (up to isomorphism) of the graph having metric dimension 2.
1. Introduction
In algebraic combinatorics, the study of graphs associated with algebraic objects is one of the most important and fascinating fields of research. During the last couple of decades, a lot of research is carried out in this field. There are many papers on assigning graphs to rings, groups, and semigroups [1–6]. Several authors [7–13] studied different properties of these graphs including diameter, girth, domination, metric dimension, central sets, and planarity.
We start by defining some basic concept related to graph theory. A graph has a vertex set and the edge set . The cardinality of the vertex set and edge set is called the order and size of , respectively. A path in is a sequence of edges . A graph is connected if every pair of vertices is connected by a path. The distance between two vertices is denoted by and is the length of the shortest path between them. The diameter of is denoted by and is defined as the largest distance between the vertices of . Let be an ordered subset of . Then, the tuple is the representation with respect to . The vertex is said to be resolved by if for any vertex . The set is called resolving set of if distinct vertices of have distinct representations with respect to , and it is called basis of if it is a resolving set with minimal cardinality. The metric dimension of , denoted by , is the cardinality of basis. The concept of metric dimension was introduced by Slater [14] and later studied by Harary and Melter [15]. It has many applications, for example, robot navigation [16], pharmaceutical chemistry [17, 18], sonar and coast guard long range navigation [14], and combinatorial optimization [19].
Let be set of nonnegative integers. A subset is said to be numerical semigroup if the following holds:(1)(2) for all (3) is finite
It is easy to observe that the numerical semigroup is a commutative monoid. Thus, the set of numerical semigroups classifies the set of all submonoids of . The elements of the set are called gaps of , and the largest element of this set is known as Frobenius number. Note that every numerical semigroup is finitely generated; that is, there exist a set such that . Moreover, every numerical semigroup has a unique minimal system of generators. The cardinality of the minimal system of generators is called embedding dimension of . It is denoted by . A subset of numerical semigroup is ideal (integral ideal) of if for all and and the element . An ideal is called irreducibly ideal if it cannot be written as intersections of two or more than two ideals which contained it properly. For more details on theory of numerical semigroup, the interested readers can refer to [20].
Recently, several authors studied the metric dimension of the graphs associated with the algebraic objects. Soleymanivarniab et al. [21] gave some metric dimension formula for annihilator graphs. Bailey et al. [22] studied the constructions of resolving sets of Kneser and Johnson graphs and provided bounds on their metric dimension. Faisal et al. [23] studied the metric dimension of the commuting graph of a dihedral group. The metric dimension of a zero-divisor graph of a commutative ring was studied in [13], while the metric dimension of a total graph of a finite commutative ring was studied in [24]. For more results on the metric dimension, we refer the readers to [25–30].
2. Notation and Preliminaries
Let be a numerical semigroup, where is the minimal system of generators of . Then, every has a representation of the form , where are nonnegative integers. Let be a fixed integer. We say that an element has a -representation if there exist and positive integers such that ; that is, can be written as linear combination of exactly generator of . Let denote the set containing all the elements , which have a representation. It is easy to see that
Note that an element may have more than one representations. For an element , we use the notation if it has a unique representation and if it has number of representations. Let , then there exist two -tuples, the coefficients tuple , and the generators -tuple such that . We denote the coefficient and generators tuple of an element by and , respectively. Also, the -th component of and is denoted by and , respectively. By using the above notations, for any , we define
For a -representation , we set
Lemma 1. With the notations defined above, we have
Proof. The proof of this lemma follows from the definition of .
Let be a numerical semigroup and be irreducible ideal of . Binyamin et al. [31] assigned a graph to numerical semigroup and studied its properties. Peng Xu et al. [32] assign a graph to the ideal of numerical semigroup with vertex set and two vertices are adjacent if and only if . Barucci [33] showed that every irreducible ideal of numerical semigroup can be expressed in the form , where , for some . Hence, the vertex set of the graph is the set for some . Peng Xu et al. [32] proved that the graph is always connected and diameter 2. The aim of this paper is to find all the graphs having metric dimension 2. The following result by Chartrand et al. [18] gives bound on the order of graph with given metric dimension and diameter .
Theorem 1. Let be a graph with metric dimension and . Let be the diameter of . Then, .
Hence, to find graphs with metric dimension 2, it is enough to classify all graphs of order less than or equal to 6. In the next section, we give bounds for the graphs of orders 4 and 5.
2.1. Bounds for the Graphs of Orders 4 and 5
Lemma 2. Let be a numerical semigroup of embedding dimension . Then, , if one of the following holds:(1) for some .(2).(3).(4) and .(5) and .
Proof. (1)If for some , then there is a -representation of in . This gives . This implies .(2)If , then there are with . We assume that and then . This gives , and therefore, .(3)If , then we have with . One can easily see that must contain , , , , and . Therefore, .(4) Lemma 1: If and then there is the unique 2-representation of . Now if for some then from (1), it follows that , and if for all , then gives . So if , then ; otherwise, . Consequently, .(5)If and , then we can assume and . This gives are in , and therefore, .
Lemma 3. Let be a numerical semigroup of embedding dimension . Then, , if one of the following holds:(1) for some .(2).(3).(4) and .
Proof. This lemma can be proved in a similar way as we proved Lemma 2.
2.2. Computation of Irreducible Ideals for the Graphs of Orders 4 and 5
Lemma 4. Let be a numerical semigroup of embedding dimension . If , then is one of the following:(1).(2) and .(3) and .
Proof. If , then from Lemma 2, it follows that satisfies one of the following conditions: and , . , and , .If and , , then has a unique 1-representation . By Lemma 1, we get . As , it follows that . This gives case (1).
Now if and , , then there are exactly two 1-representations, say and of . Assume that , then and is not a multiple of . Then, it follows from Lemma 1 thatWe show that . Let for some with . Then, , and we get . This gives , a contradiction. Therefore, we have . As , and is the only possibility. This gives case (2).
Let and , . Then, we can assume and are the only possible 1-representation and 2-representation of , respectively. By (2) in Lemma 2, we have . In this case, it is easy to see that . Then, gives and we get case (3).
Lemma 5. Let be a numerical semigroup of embedding dimension . If , then is one of the following:(1).(2).(3) and .(4) and .
Proof. Given that , then from Lemma 5, it follows that satisfies one of the following conditions: and , . and , . , and , .These possibilities can be checked in a similar way as we did in Lemma 4 to get the required result.
3. Graphs with Metric Dimension 2
Theorem 2. There are exactly 5 nonisomorphic graphs with metric dimension 2.
We prove Theorem 2 in a sequence of following lemmas.
Lemma 6. There are exactly 2 nonisomorphic graphs with 4 or less vertices and metric dimension 2.
Proof. It is trivial to note that no such graph exists for .
Now if , then from Lemma 4, we have the following possibilities:(1) with , .(2) with , .(3) with , .If (1) holds, then , and therefore, is isomorphic to the graph given in Figure 1. So metric dimension of is 2.
Now if (2) or (3) holds, then either or . In both cases, is isomorphic to the graph given in Figure 2, and therefore, metric dimension of is 2.
... There are many papers on assigning graphs to rings, groups and semigroups see [1,2,3,4,5,6]. Several authors [7,8,9,10,11] studied different properties of these graphs including diameter, girth, domination, central sets and planarity. Let Λ be a numerical semigroup. ...
Let $\Lambda$ be a numerical semigroup and $I\subset \Lambda$ be an ideal of $\Lambda$. The graph $G_I(\Lambda)$ assigned to an ideal $I$ of $\Lambda$ is a graph with elements of $(\Lambda \setminus I)^*$ as vertices and any two vertices $x,y$ are adjacent if and only if $x+y \in I$. In this paper we give a complete characterization (up to isomorphism ) of the graph $G_I(\Lambda)$ to be planar, where $I$ is an irreducible ideal of $\Lambda$. This will finally characterize non planar graphs $G_I(\Lambda)$ corresponding to irreducible ideal $I$.
... AG(R) is a graph whose vertices are ideals of R with nonzero annihilators and in which two vertices I and J are adjacent if and only if IJ = (0). Later, it was modified and further studied by many authors (see [1,2,3,18,20]). ...
Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(\tau_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.
... The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A * (R) and two distinct vertices I 1 and I 2 are adjacent if and only if I 1 I 2 = (0). In the literature, there are many papers assigning graphs to ideals of rings, see [10][11][12][13][14]. In [15], Eslahchi and Rahimi have introduced and investigated a graph called the k-zero-divisor hypergraph of a commutative ring. ...
The concept of the annihilating-ideal graph of a commutative ring was introduced by Behboodi et. al in 2011. In this paper, we extend this concept to the hypergraph for which we define an algebraic structure called k-annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let R be a commutative ring and k an integer greater than 2 and let A(R,k) be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by AG k (R), is a hypergraph with vertex set A(R,k), and for distinct elements I 1 ,I 2 ,…,I k in A(R,k), the set {I 1 ,I 2 ,…,I k } is an edge of AG k (R) if and only if ∏i=1kI i =(0) and the product of any (k−1) elements of the {I 1 ,I 2 ,…,I k } is nonzero. In this paper, we provide a necessary and sufficient condition for the completeness of 3-annihilating-ideal hypergraph of a commutative ring. Further, we study the planarity of AG 3 (R) and characterize all commutative ring R whose 3-annihilating-ideal hypergraph AG 3 (R) is planar.
... Behboodi and Rakeei [14,15] have introduced and investigated the annihilating-ideal hypergraph of a commutative ring. In the literature, there are many papers assigning graphs to ideals of rings, see [1,2,14,15,28,30,32]. From this view, Selvakumar et al. [26] introduced a hypergraph called k-annihilating-ideal hypergraph of a commutative ring R. For a commutative ring R and k ≥ 2 a fixed integer, a nonzero proper ideal I 1 in R is said to be a k-annihilating-ideal in R if there exist elements of {I 1 , I 2 , . . . ...
Let R be a commutative ring and k an integer greater than 2 and let A(R, k)
be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph
of R, denoted by AGk(R), is a hypergraph with vertex set A(R, k), and for
distinct elements I1, I2, . . . , Ik in A(R, k), the set {I1, I2, . . . , Ik} is an edge of
AGk(R) if and only if the product of these k ideal is zero and the product of any (k − 1) elements of the {I1, I2, . . . , Ik} is nonzero. In this paper, we characterize all Artinian
commutative nonlocal rings R whose AG3(R) has genus one.
Let \(\Lambda \) be a numerical semigroup and \(I\subset \Lambda \) be an ideal of \(\Lambda \). The graph \(G_I(\Lambda )\) assigned to an ideal I of \(\Lambda \) is a graph with elements of \((\Lambda {\setminus } I)^*\) as vertices and any two vertices x, y are adjacent if and only if \(x+y \in I\). In this paper we give a complete characterization (up to isomorphism ) of the graph \(G_I(\Lambda )\) to be planar, where I is an irreducible ideal of \(\Lambda \). This will finally characterize non-planar graphs \(G_I(\Lambda )\) corresponding to irreducible ideal I.
Let R be a commutative ring with identity and Nil(R) be the ideal consisting of all
nilpotent elements of R. Let I(R) = {I : I is a non-trivial ideal of R and there exists
a non-trivial ideal J such that IJ ⊆ N il(R)}. The nil-graph of ideals of R is defined
as the graph AGN (R) whose vertex set is the set I(R) and two distinct vertices I and
J are adjacent if and only if IJ ⊆ Nil(R). In this paper, we discuss some properties of
nil-graph of ideals concerning connectedness, split and claw free. Also we characterize
all commutative Artinian rings R for which the nil-graph AGN (R) has genus two
Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with
nonzero annihilator. In this paper and its sequel, we introduce and investigate
the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is
the (undirected) graph with vertices
${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$
and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness
conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a
domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if
$R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of
${\Bbb{AG}}(R)$ 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 $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$
has only $n$ nonzero proper ideals. Next, we study the connectivity of
${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and
$diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then
$gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$
is complete or star, are characterized, as well as rings $R$ for which every
vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall
study the diameter and coloring of annihilating-ideal graphs.
For each commutative ring R we associate a (simple) graph Γ(R). We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R).
For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R 2 × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.
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