Some Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings

Abstract

Let R be a commutative finite principal ideal ring with unity, and let G(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull number for this graph. We also determine when G(R) is chordal. Finally, we study some properties of the complement graph of G(R).

1. Introduction

All rings are assumed to be finite commutative principal ideal rings with unity .

For each vertex in a graph , let be the number of vertices adjacent to and let be the set of vertices adjacent to in . For any undefined graph theoretical terms, the reader may consult [1].

Let be a family of nonempty sets. The intersection graph defined on is a simple graph whose vertex set is and two vertices and are adjacent if and . Many authors worked on the graphs when the members of have an algebraic structure; see, for example, [2–7].

The intersection graph of ideals of a ring is a simple graph whose vertices are the nontrivial proper ideals and two vertices are adjacent if and . Note that if is a field, then is the null graph which has no vertices. Extending statements to the null graph would introduce unnecessary distractions, so we ignore the null graph; that is, all rings are assumed to be nonfields except when stated explicitly.

In this paper we consider the intersection graph of nontrivial proper ideals of a finite commutative principal ideal ring with unity . If is a finite commutative ring, then it can be written as a product of local rings; see [8]. If and has a unity, then any ideal of can be written as , where is an ideal in for each , while if has no unity, then this needs not to be true; see [9, page 135]. If is a finite local principal ideal ring with maximal ideal , then the ideals of are , , and for some ; see the proof of Proposition 8.8 in [10].

This study is a continuation of the study in [2], where the author used the fact that if is a local ring with maximal ideal , then there exists such that but , to define . If is a nonlocal ring such that , where is a local ring with for , then has nontrivial proper ideals. The author characterized when is Eulerian, Hamiltonian, planar, or bipartite.

In this paper we will continue the investigation of properties of the intersection graph and calculate the radius, eccentricity, dominating number, independence number, geodetic number, and the hull number; we also determine when is chordal. We conclude this paper by a study of some properties of the complement graph of .

2. The -Cube

The -cube    is the graph whose vertex set is the set of all binary -tuples, where two -tuples are adjacent if and only if they differ in precisely one coordinate.

Theorem 1. Let , where is a finite local principal ideal ring for each . Then contains a subgraph isomorphic to . In particular, if each is a field, then this subgraph is a spanning subgraph of .

Proof. Let . Consider the one-to-one correspondence , defined by , where If and are two adjacent vertices of , then there exists a unique such that ; say and . Since , we can find such that . But , and hence and are adjacent in because . Therefore the subgraph of induced by contains a copy of . Note that the set consists of all vertices of when is a product of fields.

The previous result shows that contains as a subgraph. This subgraph is induced only when . If , where and is a local ring for each , then and , where , and for and , , and for are two elements of . Clearly, and are adjacent in the subgraph of induced by , while and are not adjacent in .

3. Dominating Sets and Numbers of

In a graph , a dominating set is a set of vertices such that every vertex outside is adjacent to at least one vertex in . The domination number of a graph , denoted by , is the smallest number of the form , where is a dominating set.

If is a finite local principal ideal ring with maximal ideal , then is complete, and so is a dominating set, and . Assume that is nonlocal and , where is local for each . We have two cases.

Case 1. One factor, say , is not a field, with maximal ideal . Let . Then is a dominating set, and .

Case 2. is a product of fields. Let be a nontrivial proper ideal in . Then is of the form , where for at least one . Let such that and for . Then , and so, cannot be a dominating set. Thus . Now, let and . Then is a dominating set, and . Thus we have the following result.

Theorem 2. For any finite principal ideal ring which is not a field, except when is a product of fields; one has .

4. Radius and Center of

For a graph , the eccentricity of a vertex in is is a vertex in . A center of is a vertex with smallest eccentricity. The eccentricity is called the radius of and is denoted by .

Now, we calculate the radius of .

Theorem 3. For any finite principal ideal ring which is not a field, one has

Proof. Let be a local ring with . Then is and so .
Assume now that is local with or is not a product of fields. If is a local ring or is not a product of fields, then there exists an ideal in such that is adjacent to every other ideal in and so, . If is a product of two fields, then and are the only vertices in and they are nonadjacent. So . Let be a product of fields with , and such that . If , then is a path in . If , then , where or and , where or . Let be the least element in such that and let be the least element in such that . It is clear that . Define such that
Then is a vertex in and is a path in . Hence and . Since there is no ideal in that is adjacent to every other ideal, we have .

Note that if is a local ring with maximal ideal , then is a center for , and if , where is a local ring with maximal ideal , then is a center for , while if , where is a field for each and , then is a center for .

The following result was proved in [2] and will be used in the proof of the next theorem.

Lemma 4. For any finite principal ideal ring which is not a field, one has

The subgraph of a graph induced by the set of centers of is denoted by . The graph is called self-centered if .

Theorem 5. Let be a finite principal ideal ring. The graph is self-centered if and only if is local or a product of fields.
Moreover, if , where , is a finite principal ideal local ring for each , and there is at least one having nilpotency greater than , then the vertex set of is and , where .

Proof. The graph is self-centered when its diameter and radius are equal. Thus, by Lemma 4 and Theorem 3, is self-centered when is local or is a product of fields with . So, suppose that , where , is a local ring for each , and there is at least one having nilpotency greater than . Then, by Theorem 3, has radius . Any vertex with for each is adjacent to every other vertex of . Thus is a center of . Now let with for some be a vertex of . Take the vertex with and for every . Obviously, and are not adjacent and hence is not a center of . Thus the vertex set of is which has cardinality , where . The fact that is a complete graph follows directly from the fact that has radius .

5. Independence Number of

An independent vertex set of a graph is a set of vertices such that no two of them are adjacent in . The vertex independence number of a graph, often called simply the  independence number, is the cardinality of the largest (vertex) independent set.

Theorem 6. If , where is a finite local principal ideal ring for each , then .

Proof. If is local, then . So assume that . Suppose that is an independent set of . Suppose that , where , , and . Define , where , for and , where , for . Observe that is an independent set whose cardinality is greater than the cardinality of . So, an independent set with maximum cardinality cannot contain any element , where for two or more indices. Thus is an independent set with maximum cardinality in and .

6. Geodetic and Hull Numbers of

Let be two vertices of a connected graph . A shortest path between and is called a geodesic. The set of all vertices in that lie on a geodesic is denoted by . For any subset of , let . If , then is called a geodetic set of . The minimum cardinality of a geodetic set of is called the geodetic number of and this number is denoted by . A subset of is convex if . If is a subset of , then the convex hull of (denoted by ) is the smallest convex set in containing . If , then is called a hull set of . The smallest cardinality of a hull set of is called the hull number of and is denoted by . It is clear that for any graph . In this section, we find the geodetic and hull numbers of .

A vertex in a graph is called an extreme vertex if the subgraph induced by its neighbors is complete. The set of extreme vertices of is denoted by . The following lemma can be found in [11] or [12].

Lemma 7. Every geodetic set (resp., hull set) of contains .

Note that if is a local ring with , then is complete, and so . The following lemma determines the set of extreme vertices of , where is a nonlocal ring.

Lemma 8. Let be a product of finite local principal ideal rings, and . Then is the set of extreme vertices of .

Proof. Let and be two vertices that are adjacent to , where for only. Then and . Thus and are adjacent. Hence the subgraph induced by the neighbors of is a complete subgraph and is an extreme vertex. Let , where , , with . Define , where , for and , where , for . Observe that and are not adjacent, but both are adjacent to . So, is not an extreme vertex of . Thus, is the set of extreme vertices of .

Theorem 9. Let be a product of finite local principal ideal rings. Then the set is a geodetic set of and , where .

Proof. Let be any vertex of . Then there are and , where , , with . Take , where and . Then is a geodesic that contains . Use Lemmas 7 and 8 to get the result.

According to Theorem 9 and Lemmas 7 and 8, the hull number of is equal to the geodetic number of . Also the set is a hull set of . We state that in the following corollary.

Corollary 10. Let be a product of finite local principal ideal rings. Then the set is a hull set of and , where .

7. When Is Chordal?

A graph is chordal if it has no induced cycle of length greater than 3.

Theorem 11. Let be a finite principal ideal ring which is not a field. The graph is chordal if and only if is the product of at most three local rings.

Proof. If , where and is a local ring for each , then the four vertices , , , and induce a -cycle . Thus is not chordal. If is local with nilpotency , then which is chordal. So, suppose that , where and is a local ring for each . Assume to the contrary that is an induced cycle in of length greater than . Since is adjacent to the two nonadjacent vertices and , we must have two different values and such that for , , and . Since is not adjacent to and we have at most three factors, we must have for . Similarly, because is adjacent to the two nonadjacent vertices and , we must have , for , for , and . But now, the other neighbor of in this cycle must be adjacent to (they have nontrivial intersection in the factor ). This contradicts the assumption that this cycle is induced in . Therefore is chordal.

8. The Complement of

The complement of a graph is the graph on the same vertex set as but two vertices are adjacent in if and only if they are nonadjacent in .

Theorem 12. Let be a finite principal ideal ring which is not a field. The graph is connected if and only if is either local with nilpotency or a product of fields.

Proof. If is local with nilpotency , then . Thus is connected if and only if (note that, for , ). If , where , is a local ring for each , and there exists such that has nilpotency greater than , then the vertex with and for is adjacent to every other vertex of .Thus is an isolated vertex in . Therefore is disconnected (note that the order of is greater than ). Finally, suppose that , where and is a field for each . If , then and hence is connected. So, assume that . Let and be two distinct nonadjacent vertices of . Then and are adjacent in . Thus there exists such that . But since , there exists such that ; say and . Take the vertex with and for . Clearly, is adjacent in to but not to . But there exists such that (since ). Now take the vertex with and for . Then is adjacent in to both and . Thus is an path in . Therefore is connected.

The next two results determine the diameter and radius of .

The diameter of a graph and its complement are some times related; for instance, if a graph is disconnected, and so , then its complement is connected with , while if is connected with , then is connected and . One now can compare Lemma 4 concerning with the next theorem.

Theorem 13. Let be a finite principal ideal ring which is not a field. Then

Proof. If is neither local with nilpotency nor a product of fields, then, by Theorem 12, we have that is disconnected and hence has infinite diameter. If is local with nilpotency or a product of two fields, then or , respectively. So, let , where and is a field for each . For any two nonadjacent vertices of we have found in the proof of Theorem 12 a path of length joining them. Thus . Take the two nonadjacent vertices and of with , for , , and for . Then the vertex with and for is the unique neighbor of in . Similarly, the vertex with and for is the unique neighbor of in . Thus and have no common neighbor and hence . Therefore, since , we have .

In Theorem 3, the radius of was calculated, and here we calculate .

Theorem 14. Let be a finite principal ideal ring which is not a field. Then

Proof. If is neither local with nilpotency nor a product of fields, then, by Theorem 12, we have that is disconnected and hence has infinite radius. If is local with nilpotency or a product of two fields, then or , respectively. So, let , where and is a field for each . Then, since by Theorem 13, we have or . To conclude that , it is enough to find a vertex with eccentricity . Take the vertex with and for . Let be another vertex of . If , then . So, suppose that . Then, since , there exists such that . But there must be such that . Now, take the vertex with and for . Then is a common neighbor of and . Therefore .

Now we determine .

Theorem 15. Let be a finite principal ideal ring which is not a field. Then is self-centered except when is a product of fields with . If is a product of fields with , then the vertex set of is

Proof. If is not a product of fields, then has equal diameter and radius by Theorems 13 and 14. Thus is self-centered. If is a product of two fields, then which is also self-centered. So, suppose that , where and is a field for each . By the same process as that in the proof of Theorem 14, we can show that any vertex of the form , with except for one value of , has eccentricity . Thus each such vertex is a center of . Now let with for at least two values and of . This means that for . Take the vertex with and for . Then the unique neighbor of in is with and for . Clearly, is not adjacent to in , and hence . Thus , which implies that is not a center of . Therefore the vertex set of is . But any two of the vertices of are adjacent in . Thus .

9. Coloring

A clique of a graph is a maximal complete subgraph of . The clique number of , denoted by , is the largest possible size of a maximum clique in . A proper coloring of a graph is a function that assigns a color to each vertex such that no two adjacent vertices have the same color. The chromatic number of , denoted by , is the smallest number of colors necessary to produce a proper coloring.

It is clear that if is a local ring with , then is , and so .

Theorem 16. Let be a finite principal ideal ring which is not a field, and let be a finite local principal ideal ring. Then .

Proof. The vertex set of is the disjoint union of the following five sets: Let . Then the vertices in can be colored by colors. The vertex is adjacent to all vertices from , so needs necessarily a new color . This new color can be assigned to every other vertex from , since is an independent set in . The vertex can be colored by one of the previous colors except , since the neighbors of are precisely the elements of . For each proper nontrivial ideal in , assign the color of to each of the vertices , where is a nontrivial ideal in . Note that the neighbors of in are precisely the neighbors of in . This completes coloring the vertices from . Finally, since every element of is an isolated vertex in , there is no need for any new color. Therefore, .

Corollary 17. Let , where each is a finite local principal ideal ring. Then .

Proof. For , we have . Thus . For , where both and are fields, we have . Thus . Therefore, the result follows by induction and Theorem 16.

Note that for , where each is a finite local principal ideal ring, we have by Theorem 6. Thus, by Corollary 17, .

10. Between and the Zero Divisors

After a conversation between the first author and Professor Christian Lomp (Porto University, Portugal), the latter suggested that there may be a relation between the graph and the zero divisor graph , since if , then clearly . In fact we manage to find a very nice result but with the graph .

In [13] the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring was introduced as follows.

Let be the set of zero divisor elements in and let . For , if . This relation is an equivalence relation, and a well-defined multiplication was defined on the set of equivalence classes of ; that is, if denotes the class of , then the product , and they note that and and the other equivalence classes form a partition of . The authors defined the graph of equivalence classes of elements of and with pair of distinct classes , joined by an edge if and only if .

Theorem 18. If is a reduced finite principal ideal ring, then .

Proof. Since is reduced it is a product of fields and so for any two ideals and of , . Thus we have the following for each .
is adjacent to in  , ,  ,   is adjacent to in .
Thus if we define such that , then is an isomorphism.

If is not reduced in the above theorem, then the result needs not to be true as one can see that .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.