On the intersection graph of ideals of a commutative ring
^{1}
Abstract
Let be a commutative ring and be an module, and let be the set of all nontrivial ideals of . The intersection graph of ideals of , denoted by , is a graph with the vertex set , and two distinct vertices and are adjacent if and only if . For every multiplication module , the diameter and the girth of are determined. Among other results, we prove that if is a faithful module and the clique number of is finite, then is a semilocal ring. We denote the intersection graph of ideals of the ring by , where are integers and is a module. We determine the values of and for which is perfect. Furthermore, we derive a sufficient condition for to be weakly perfect.
1 Introduction
Let be a commutative ring, and be the set of all nontrivial ideals of .
There are many papers on assigning a graph to a ring , for instance see [1–4].
Also the intersection graphs of some algebraic structures such as groups, rings and modules have been studied by several authors, see [3, 6, 8].
In [6], the intersection graph of ideals of , denoted by , was introduced as the graph with vertices and for distinct ,
the vertices and are adjacent if and only if . Also in [3], the intersection
graph of submodules of an module , denoted by , is defined to be the graph whose vertices are
the nontrivial submodules of and two distinct vertices are
adjacent if and only if they have nonzero intersection.
In this paper, we generalize to , the intersection graph of ideals of , where is an module.
Throughout the paper, all rings are commutative with nonzero identity and all modules are unitary.
A module is called a uniform module if the intersection of any two nonzero submodules is nonzero. An module is said to be a multiplication module if every submodule of is of the form , for some ideal of .
The annihilator of is denoted by . The module is called a faithful module if . By a nontrivial submodule of
, we mean a nonzero proper submodule of . Also, denotes
the Jacobson radical of and denotes the ideal of all nilpotent elements of . By , we denote the set of all maximal ideals of .
A ring having only finitely many maximal ideals is said to be a semilocal ring.
As usual, and will denote the integers and the integers modulo , respectively.
A graph in which any two distinct vertices are adjacent is called a complete graph. We denote the complete graph on vertices by . A null graph is a graph containing no edges. Let be a graph. The complement of is denoted by . The set of vertices and the set of edges of are
denoted by and , respectively. A subgraph of is said to be an induced subgraph of if it has exactly the edges that appear
in over . Also, a subgraph
of is called a spanning subgraph if . Suppose that . We denote by the degree of a vertex in . A regular graph is a graph where each vertex has the same degree. We recall
that a walk between and is a sequence — — — of vertices of such that for every with ,
the vertices and are adjacent. A path between and is a walk between and without repeated vertices. We say that is
connected if there is a path between any two distinct
vertices of .
For vertices and of , let be the length of a shortest path from to ( and if there is no path between
and ). The diameter of , , is the supremum of the set .
The girth of , denoted by , is the length of a shortest cycle in ( if contains no cycles).
A clique in is a set of pairwise adjacent vertices and the number of vertices in the largest clique of ,
denoted by , is called the clique number of . The chromatic number of , , is the minimal number of colors which can be assigned to the vertices of in such a way that every two adjacent vertices have different colors. A graph is perfect if for every induced subgraph of , . Also, is called weakly perfect if .
In the next section, we introduce the intersection graph of ideals of , denoted by , where is a commutative ring and is a
nonzero module. It is shown that for every multiplication module , and . Among other results, we prove that if is a faithful module and
is finite, then and . In the last section, we consider the intersection graph of ideals of , denoted by , where are integers and is a module. We show that is a perfect graph if and only if has at most four distinct prime divisors. Furthermore, we derive a sufficient condition for to be weakly perfect. As a corollary, it is shown that the intersection graph of ideals of is weakly perfect, for every integer .
2 The intersection graph of ideals of
In this section, we introduce the intersection graph of ideals of and study its basic properties.
Definition.
Let be a commutative ring and be a nonzero module. The intersection graph of ideals of , denoted by , is the
graph with vertices and two distinct vertices and are adjacent if and only if .
Clearly, if is regarded as a module over itself, that is, , then the intersection graph of ideals of is exactly the same as the intersection graph of ideals of . Also, if and are two isomorphic modules, then is the same as .
Example 1
. Let . Then we have the following graphs.
{tikzpicture}\GraphInit[vstyle=Classic] \Vertex[x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]4 \Vertex[x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]2 \Vertex[x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]6 \Vertex[x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]3 \Edges(2,3) \Edges(6,3) \Edges(2,6) \Edges(2,4)
{tikzpicture}\GraphInit[vstyle=Classic] \Vertex[x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]4 \Vertex[x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]2 \Vertex[x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]6 \Vertex[x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]3
{tikzpicture}\GraphInit[vstyle=Classic] \Vertex[x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]4 \Vertex[x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]2 \Vertex[x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]6 \Vertex[x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]3 \Edges(2,4)
{tikzpicture}\GraphInit[vstyle=Classic] \Vertex[x=1,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]4 \Vertex[x=1,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]2 \Vertex[x=2.3,y=0,style=black,minimum size=3pt,LabelOut=true,Lpos=270,L=]6 \Vertex[x=2.3,y=1,style=black,minimum size=3pt,LabelOut=true,Lpos=90,L=]3 \Edges(2,3) \Edges(6,3) \Edges(2,6)
Example 2
. Let be an integer. If is the least common multiple of two distinct integers , then . Thus and are adjacent in if and only if does not divide .
Example 3
. Let be a prime number and be two positive integers. If divides , then is an isolated vertex of . Therefore, since is a uniform module, so is a disjoint union of an infinite complete graph and its complement. Also, (the quasicyclic group), is a uniform module and . Hence is an infinite complete graph.
Remark 1
. Obviously, if is a faithful multiplication module, then is a complete graph if and only if is a uniform module.
Remark 2
. Let be a commutative ring and let be a nonzero module.

If is a faithful module, then is a spanning subgraph of . To see this, suppose that and are adjacent vertices of . Then implies that and so . Therefore is adjacent to in .

If is a multiplication module, then is an induced subgraph of . Note that for each nontrivial submodule of , there is a nontrivial ideal of , such that and so we can assign to . Also, is adjacent to in if and only if , that is, if and only if is adjacent to in .
Theorem 1
. Let be a commutative ring and let be a faithful module. If is not connected, then is a direct sum of two modules.
Proof.
Suppose that and are two distinct components of . Let and . Since is a faithful module, so implies that and . Now if , then — — is a path between and , a contradiction. Thus and so .
The next theorem shows that for every multiplication module , the diameter of has possibilities.
Theorem 2
. Let be a commutative ring and be a multiplication module. Then .
Proof.
Assume that is a connected graph with at least two vertices. So is a faithful module. If there is a nontrivial ideal of such that , then is adjacent to all other vertices. Hence . Otherwise, we claim that is connected. Let and be two distinct vertices of . Since is a multiplication module, so and , for some nontrivial ideals and of . Suppose that — — — is a path between and in . Therefore, — — — — is a walk between and . Thus, we conclude that there is also a path between and in . The claim is proved. So by [3, Theorem 2.4], . Now, suppose that and are two distinct vertices of . If , then and are two distinct vertices of . Hence there exists a nontrivial submodule of which is adjacent to both and in . Since is a multiplication module, so , for some nontrivial ideal of . Thus is adjacent to both and in . Therefore .
Theorem 3
. Let be a commutative ring and be a multiplication module. If is a connected regular graph of finite degree, then is a complete graph.
Proof.
Suppose that is a connected regular graph of finite degree. If , then . So assume that . We claim that is an Artinian module. Suppose to the contrary that is not an Artinian module. Then there is a descending chain of submodules of , where ’s are nontrivial ideals of . This implies that is infinite, a contradiction. The claim is proved. Therefore has at least one minimal submodule. To complete the proof, it suffices to show that contains a unique minimal submodule. By contrary, suppose that and are two distinct minimal submodules of . Hence and , where and are two nontrivial ideals of . Since , so and are not adjacent. By Theorem 2, there is a vertex which is adjacent to both and . So both and are contained in . Thus each vertex adjacent to is adjacent to too. This implies that , a contradiction.
Also, the following theorem shows that for every multiplication module , the girth of has possibilities.
Theorem 4
. Let be a commutative ring and be a multiplication module. Then .
Proof.
Suppose that — — — — is a cycle of length in . If , we are done. Thus assume that . Since and is a multiplication module, we have , where is a nonzero ideal of . If is a proper ideal of and , then — — — is a triangle in . Otherwise, we conclude that or . Similarly, we can assume that or , for every , . Without loss of generality suppose that . Now, if , then — — — is a cycle of length 3 in . Therefore assume that . Since or , so — — — is a triangle in . Hence if contains a cycle, then .
Lemma 1
. Let be a commutative ring and be a nonzero module. If is an isolated vertex of , then the following hold:

is a maximal ideal of or .

If , then , for every .
Proof.
There is a maximal ideal of such that . Assume that . Then we have
, since is an isolated vertex. So .
Suppose that and . Since is an isolated vertex, we have and so , a contradiction.
Thus .
Theorem 5
. Let be a commutative ring and be a faithful module. If is a null graph, then it has at most two vertices and is isomorphic to one of the following rings:

, where and are fields;

, where is a field;

, where is a coefficient ring of characteristic , for some prime number .
Proof.
In the next theorem we show that if is a faithful module and , then is a semilocal ring.
Theorem 6
. Let be a commutative ring and be a faithful module. If is finite then and .
Proof.
First we prove that . Let . By contradiction, assume that
are distinct maximal ideals of . We know that , for every ,
. Otherwise, , for some , . So
and hence by Prime Avoidance Theorem [5, Proposition 1.11],
we have , for some , , which is impossible. This implies that
is a clique in , a contradiction.
Thus .
Now, we prove that . By contrary, suppose that . Since , for every , and is finite,
we conclude that , for some integers . Hence , for some . Since , so is a unit.
This yields that , a contradiction. The proof is complete.
3 The intersection graph of ideals of
Let be two integers and be a module. In this section we study the intersection graph of ideals of the ring . Also, we generalize some results given in [9]. For abbreviation, we denote by .
Clearly, is a module if and only if divides .
Throughout this section, without loss of generality, we assume that and , where ’s are distinct primes, ’s are positive integers, ’s are nonnegative integers, and for . Let and . The cardinality of is denoted by .
For two integers and , we write () if divides ( does not divide ).
First we have the following remarks.
Remark 3
. It is easy to see that divides and . Let be a module. If , then is an isolated vertex of . Obviously, and are adjacent if and only if . This implies that is a subgraph of .
Remark 4
. Let be a module and be a divisor of . We set . Clearly, . Suppose that is a clique of . Then is an intersecting family of subsets of . (A family of sets is intersecting if any two of its sets have a nonempty intersection.) Also, if is an intersecting family of subsets of and is nonempty, then is a clique of . (If is a nonempty subset of and , then we will denote by .) Thus we have
Now, we provide a lower bound for the clique number of .
Theorem 7
. Let be a module. Then
Proof.
Suppose that . With the notations of the previous remark, let . Then is an intersecting family of subsets of and so is a clique of . Clearly, . Therefore and hence the result holds.
Clearly, if , then equality holds in the previous theorem. Also, if has only two distinct prime divisors, that is, , then again equality holds. So the lower bound is sharp.
Example 4
. Let , where are distinct primes. Thus and . It is easy to see that and . Also, . Let , for . Hence , for . If , then . Therefore .
By the strong perfect graph theorem, we determine the values of and for which is a perfect graph.
Theorem A
. (The Strong Perfect Graph Theorem [7]) A finite graph is perfect if and only if neither nor contains an induced odd cycle of length at least .
Theorem 8
. Let be a module. Then is perfect if and only if has at most four distinct prime divisors.
Proof.
First suppose that and , where ’s are distinct primes and ’s are positive integers.
Let , , , , and . Now, assume that , for . Hence — — — — — is an induced cycle of length 5 in . So by Theorem A, is not a perfect graph.
Conversely, suppose that is not a perfect graph. Then by Theorem A, we have the following cases:
Case 1. — — — — — is an induced cycle of length 5 in . Let , for . So and , for . Let and , for . Clearly, are distinct and thus .
Case 2. — — — — — is an induced path of length 5 in . Let , for . So , for . Let , for . Clearly, are distinct and hence .
Case 3. There is an induced cycle of length 5 in . So contains an induced cycle of length 5 and by Case 1, we are done.
Case 4. — — — — — is an induced path of length 5 in . Since , and , we may assume that , where and , for some distinct . Similarly, we find that , for some distinct and also . Now, since and , we deduce that .
Corollary 1
. The graph is perfect if and only if has at most four distinct prime divisors.
In the next theorem, we derive a sufficient condition for to be weakly perfect.
Theorem 9
. Let be a module. If for each , then is weakly perfect.
Proof.
Let be a nonempty subset of and . As we mentioned in Remark 4, if is nonempty, then is a clique of . Also, the vertices of (if ) are adjacent to all nonisolated vertices. Suppose that and are two nonempty subsets of and . Since for each , so . This implies that and hence .
Let be an intersecting family of subsets of and .
Let . We show that or . Assume that . So there is such that . Thus and hence .
We claim that , for each . Suppose to the contrary, and . If and , then . So we have . Let . Then is an intersecting family of subsets of and , a contradiction. The claim is proved.
Now, we show that has a proper vertex coloring. First we color all vertices of with different colors.
Next we color each family of vertices out of with colors of vertices of . Note that if , then and . Suppose that and are two adjacent vertices of . Thus .
Without loss of generality, one can assume
. So we deduce that and .
Therefore, and have different colors. Thus and hence .
As an immediate consequence of the previous theorem, we have the next result.
Corollary 2
. The graph is weakly perfect, for every integer .
In the case that for each , we determine the exact value of . It is exactly the lower bound obtained in the Theorem 7.
Theorem 10
. Let be a module. If for each , then .
Proof.
Let be a proper subset of . Then and hence . Also, the vertices of (if ) are adjacent to all nonisolated vertices and . Clearly if is an intersecting family of subsets of , then . Moreover, if and , then . Thus by Theorem 9, .
Corollary 3
. Let , where ’s are distinct primes. Then .
We close this article by the following problem.
Problem. Let be a module. Then is it true that is a weakly perfect graph?
Footnotes
 thanks: Keywords: Intersection graph, perfect graph, clique number, chromatic number, diameter, girth.
2010 Mathematics Subject Classification: 05c15, 05c17, 05c69, 13a99, 13c99.
References
 S. Akbari, F. Heydari, The regular graph of a noncommutative ring, Bull. Aust. Math. Soc., 89 (2014), 132–140.
 S. Akbari, S. Khojasteh, Commutative rings whose cozerodivisor graphs are unicyclic or of bounded degree, Comm. Algebra, 42 (2014), 1594–1605.
 S. Akbari, H. A. Tavallaee, S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl., 11 (2012), Article No. 1250019.
 D. F. Anderson, A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706–2719.
 M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, AddisonWesley Publishing Company, 1969.
 I. Chakrabarty, S. Ghosh, T. K. Mukherjee, M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 (2009), 5381–5392.
 M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math., 164 (2006), 51–229.
 B. Csákány, G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J., 19 (1969), 241–247.
 R. Nikandish, M. J. Nikmehr, The intersection graph of ideals of is weakly perfect, Utilitas Mathematica, to appear.
 F. I. Perticani, Commutative rings in which every proper ideal is maximal, Fund. Math., 71 (1971), 193–198.
 J. Reineke, Commutative rings in which every proper ideal is maximal, Fund. Math., 97 (1977), 229–231.