On the intersection graph of ideals of a commutative ring Keywords: Intersection graph, perfect graph, clique number, chromatic number, diameter, girth.   2010 Mathematics Subject Classification: 05C15, 05C17, 05C69, 13A99, 13C99.

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 non-trivial 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 non-trivial 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 non-trivial submodules of and two distinct vertices are adjacent if and only if they have non-zero 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 non-zero identity and all modules are unitary. A module is called a uniform module if the intersection of any two non-zero submodules is non-zero. 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 non-trivial submodule of , we mean a non-zero 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 non-zero -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 non-zero -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 quasi-cyclic -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 non-zero -module.

  1. 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 .

  2. If is a multiplication -module, then is an induced subgraph of . Note that for each non-trivial submodule of , there is a non-trivial 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 non-trivial 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 non-trivial 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 non-trivial submodule of which is adjacent to both and in . Since is a multiplication module, so , for some non-trivial 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 non-trivial 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 non-trivial 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 non-zero 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 non-zero -module. If is an isolated vertex of , then the following hold:

  1. is a maximal ideal of or .

  2. 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:

  1. , where and are fields;

  2. , where is a field;

  3. , where is a coefficient ring of characteristic , for some prime number .

Proof.

By Lemma 1, every non-trivial ideal of is maximal and so by [10, Theorem 1.1], cannot have more than two different non-trivial ideals. Thus has at most two vertices. Also, by [11, Theorem 4], is isomorphic to one of the mentioned rings.

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 non-negative 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 non-empty intersection.) Also, if is an intersecting family of subsets of and is non-empty, then is a clique of . (If is a non-empty 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 non-empty subset of and . As we mentioned in Remark 4, if is non-empty, then is a clique of . Also, the vertices of (if ) are adjacent to all non-isolated vertices. Suppose that and are two non-empty 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 non-isolated 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

  1. thanks: Keywords: Intersection graph, perfect graph, clique number, chromatic number, diameter, girth.
      2010 Mathematics Subject Classification: 05c15, 05c17, 05c69, 13a99, 13c99.

References

  1. S. Akbari, F. Heydari, The regular graph of a noncommutative ring, Bull. Aust. Math. Soc., 89 (2014), 132–140.
  2. S. Akbari, S. Khojasteh, Commutative rings whose cozero-divisor graphs are unicyclic or of bounded degree, Comm. Algebra, 42 (2014), 1594–1605.
  3. S. Akbari, H. A. Tavallaee, S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl., 11 (2012), Article No. 1250019.
  4. D. F. Anderson, A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706–2719.
  5. M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
  6. I. Chakrabarty, S. Ghosh, T. K. Mukherjee, M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 (2009), 5381–5392.
  7. M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math., 164 (2006), 51–229.
  8. B. Csákány, G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J., 19 (1969), 241–247.
  9. R. Nikandish, M. J. Nikmehr, The intersection graph of ideals of is weakly perfect, Utilitas Mathematica, to appear.
  10. F. I. Perticani, Commutative rings in which every proper ideal is maximal, Fund. Math., 71 (1971), 193–198.
  11. J. Reineke, Commutative rings in which every proper ideal is maximal, Fund. Math., 97 (1977), 229–231.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...