ISRN Discrete MathematicsVolume 2011 (2011), Article ID 491936, 10 pagesdoi:10.5402/2011/491936

Research Article

The -Total Graph of an -Module

Reza Ebrahimi Atani

Department of Computer Engineering, University of Guilan, P.O. Box 3756, Rasht 41996-13769, Iran

Received 30 August 2011; Accepted 5 October 2011

Academic Editor: R. Yeh

Copyright © 2011 Reza Ebrahimi Atani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a complete lattice. We introduce and investigate the -total graph of an -module over an -commutative ring. The main purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to more generalize the -total graph of an -module case.

1. Introduction

It was Beck (see [1]) who first introduced the notion of a zero-divisor graph for commutative rings. This notion was later redefined by Anderson and Livingston in [2]. Since then, there has been a lot of interest in this subject, and various papers were published establishing different properties of these graphs as well as relations between graphs of various extensions (see [2–5]). Let be a commutative ring with being its set of zero-divisors elements. The total graph of , denoted by , is the (undirected) graph with all elements of as vertices, and, for distinct , the vertices and are adjacent if and only if . The total graph of a commutative ring has been introduced and studied by Anderson and Badawi in [3]. In [6], the notion of the total torsion element graph of a module over a commutative ring is introduced.

In [7], Zadeh introduced the concept of fuzzy set, which is a very useful tool to describe the situation in which the data is imprecise or vague. Many researchers used this concept to generalize some notions of algebra. Goguen in [8] generalized the notion of fuzzy subset of to that of an -subset, namely, a function from to a lattice . In [9], Rosenfeld considered the fuzzification of algebraic structures. Liu [10] introduced and examined the notion of a fuzzy ideal of a ring. Since then several authors have obtained interesting results on -ideals of a ring and -modules (see [11, 12]). Also, -zero-divisor graph of an -commutative ring has been introduced and studied in [13].

In the present paper we introduce a new class of graphs, called the -total torsion element graph of a -module (see Definition 2.2), and we completely characterize the structure of this graph. The total torsion element graph of a module over a commutative ring and the -total torsion element graph of a -module over a -commutative ring are different concepts. Some of our results are analogous to the results given in [6]. The corresponding results are obtained by modification, and here we give a complete description of the -total torsion element graph of an -module.

For the sake of completeness, we state some definitions and notation used throughout. For a graph , by and , we denote the set of all edges and vertices, respectively. We recall that a graph is connected if there exists a path connecting any two distinct vertices. The distance between two distinct vertices and , denoted by , is the length of the shortest path connecting them (if such a path does not exist, then and ). The diameter of a graph , denoted by , is equal to . A graph is complete if it is connected with diameter less than or equal to one. The girth of a graph , denoted , is the length of the shortest cycle in , provided contains a cycle; otherwise, . We denote the complete graph on vertices by and the complete bipartite graph on and vertices by (we allow and to be infinite cardinals). We will sometimes call a a star graph. We say that two (induced) subgraphs and of are disjoint if and have no common vertices and no vertex of (resp., ) is adjacent (in ) to any vertex not in (resp., ).

Let be a commutative ring, and stands for a complete lattice with least element 0 and greatest element 1. By an -subset of a nonempty set , we mean a function from to . If , then is called a fuzzy subset of . denotes the set of all -subsets of . We recall some definitions and lemmas from the book [12], which we need for development of our paper.

Definition 1.1. An -ring is a function , where is a ring, which satisfies the following.(1).(2) for every in .(3) for every in .

Definition 1.2. Let . Then is called an -ideal of if for every the following conditions are satisfied. (1).(2).The set of all -ideals of is denoted by .

Definition 1.3. Assume that is an -module, and let . Then is called an -fuzzy -module of if for all and for all the following conditions are satisfied.(1).(2).(3).The set of all -fuzzy -modules of is denoted by .

Lemma 1.4. Let be a module over a ring , and . Then for every .

2. Is a Submodule of

Let be a module over a commutative ring , and let . The structure of the -total torsion element graph may be completely described in those cases when -torsion elements form a submodule of . We begin with the key definition of this paper.

Definition 2.1. Let be a module over a commutative ring , and let . A -torsion element is an element with for which there exists a nonzero element of such that .

The set of -torsion elements in will be denoted by .

Definition 2.2. Let be a module over a ring , and let . We define the -total torsion element graph of an -module as follows: , .

Notation 1. For the -torsion element graph , we denote the diameter, the girth, and the distance between two distinct vertices and , by , , and , respectively.

Remark 2.3. Let be a module over a ring , and let . Clearly, if is a nonzero constant, then . So throughout this paper, we will assume, unless otherwise stated, that is not a nonzero constant. Thus, there is a nonzero element of such that .

We will use to denote the set of elements of that are not -torsion elements. Let be the (induced) subgraph of with vertices , and let be the (induced) subgraph of with vertices .

Definition 2.4. Let be a module over a ring , and . One defines the set by , the -annihilator of .

Lemma 2.5. Let be a module over a ring , and let . Then is an -ideal of .

Proof. Let and . If , then we have and . It then follows from Lemma 1.4 that ; hence . Similarly, .

Theorem 2.6. Let be a module over a ring and let . Then the -torsion element graph is complete if and only if .

Proof. If , then for any vertices , one has ; hence they are adjacent in . On the other hand, if is complete, then every vertex is adjacent to 0. Thus, for every . This completes the proof.

Theorem 2.7. Let be a module over a ring , and let such that is a submodule of . Then one has the following. (i) is a complete (induced) subgraph of and is disjoint from .(ii)If annμ, then is a complete graph.

Proof. (i) is complete directly from the definition. Finally, if and were adjacent, then ; so this, since is a submodule, would lead to the contradiction .(ii) Let . we may assume that . By assumption, there exists with , so . Thus , and; therefore, is a complete graph by Theorem 2.6.

Theorem 2.8. Let be a module over a ring , and let . Then is totally disconnected if and only if has characteristic 2 and .

Proof. If , then the vertices and are adjacent if and only if . Then is a disconnected graph, and its only edges are those that connect vertices and (we do not need a priori assumption that has characteristic 2). Conversely, assume that is totally disconnected. Then for every nonzero element of . Thus, . Further, since , we have (so ) for every with by the total disconnectedness of the graph . As , it follows that . Thus, .

Proposition 2.9. Let be a module over a ring , and let such that is a submodule of . If , then if and only if .

Proof. First suppose that . Since , we get that , and, for all implies that . Since , there is a nonzero element such that , and, since , one must have ; hence, . Conversely, assume that . Then there exists with . Since , we have .

Theorem 2.10. Let be a module over a ring , and let such that is a proper submodule of . Then is disconnected.

Proof. If , then is disconnected by Theorem 2.8. If , then the subgraphs of and are disjoint by Theorem 2.7 (i), as required.

Theorem 2.11. Let be a module over a ring , and let such that is a proper submodule of . Suppose and . Then one has the following. (i)If , then is a union of disjoint complete graphs .(ii)If , then is a union of disjoint bipartite graphs and one complete graph .

Proof. (i) Assume that and let be such that . The elements , from the same coset are adjacent if and only if , so , according to the Proposition 2.9. Then and are not adjacent (otherwise, we would have ), and; therefore, . Since every coset has cardinality , we conclude that is the disjoint union of complete graph . (ii) If , then the elements , from are obviously not adjacent. The elements , from different cosets are adjacent if and only if or . In this way we obtain that the subgraph spanned by the vertices from is a disjoint union of ( if is infinite) disjoint bipartite graph .

Proposition 2.12. Let be a module over a ring , and let such that is a proper submodule of . Then one has the following. (i) is complete if and only if either or .(ii) is connected if and only if either or .(iii) and, hence; ( and ) is totally disconnected if and only if and .

Proof. Let and .(i)Let be complete. Then, by Theorem 2.11, is complete if and only if is a single or . If , then . Thus, , and hence . If , then and . Thus, and ; hence, . The reverse implication may be proved in a similar way as in [6, Theorem  2.6 (1)].(ii)By theorem 2.11, is connected if and only if is a single or . Thus, either if or if ; hence, or , respectively, as needed. The reverse implication may be proved in a similar way as in [3, Theorem  2.6 (2)].(iii) is totally disconnected if and only if it is a disjoint union of ’s. So by Theorem 2.11, and , and the proof is complete.

By the proof of the Proposition 2.12, the next theorem gives a more explicit description of the diameter of .

Theorem 2.13. Let be a module over a ring , and let such that is a proper submodule of . Then one has the following. (i) if and only if and .(ii) if and only if either and or and .(iii) if and only if and .(iv)Otherwise, .

Proposition 2.14. Let be a module over a ring , and let such that is a proper submodule of . Then or . In particular, if contains a cycle.

Proof. Let contain a cycle. Then since is disjoint union of either complete or complete bipartite graphs by Theorem 2.11, it must contain either a 3 cycles or a 4 cycles. Thus .

Theorem 2.15. Let be a module over a ring , and let such that is a proper submodule of . Then one has the following. (i)(a) if and only if and .(b) if and only if and .(c)Otherwise, .(ii)(a) if and only if .(b) if and only if and .(c)Otherwise, .

Proof. Apply Theorem 2.11, Proposition 2.14, and Theorem 2.7 (i).

The previous theorems give a complete description of the structure of the -total torsion element graph of an -module when is a submodule. The question under what conditions is a submodule of and how is this related to the condition that is an ideal in naturally arises. We prove that the following results holds.

Theorem 2.16. Let be a module over a ring , and let . Then one has the following. (i)If , then is a submodule of .(ii)If is a principal ideal of with a nilpotent element of , then is a submodule of .

Proof. (i) Let and . There are nonzero elements such that , , and with (since is an integral domain). It follows that ; hence, by Lemma 1.4. Thus, . Similarly, , and this completes the proof.(ii) Assume that is not a submodule of . Then there are elements such that . By assumption, there exist nonzero elements such that , where and . Then and , so we must have , and; thus, . Since is nilpotent, we have and , for some . We may assume that . Then for the nonzero element of we have which is contrary to the assumption that .

Example 2.17. Assume that is the ring integers, and let . We define the mapping by Then and . Thus, is a complete graph by Theorem 2.6.

Example 2.18. Let denote the ring of integers modulo 8 and the ring of integers modulo 25. We define the mappings by and by Then, for each (), , , and . An inspection will show that and are submodules of and , respectively. Therefore, by Theorem 2.11, we have the following results.(1)Since , we conclude that is a union of 2 disjoint .(2)Since , we conclude that is a disjoint union of 2 complete graph and 5 bipartite .

3. Is Not a Submodule of

We continue to use the notation already established, so is a module over a commutative ring and . In this section, we study the -torsion element graph when is not a submodule of .

Lemma 3.1. Let be a module over a ring , and let such that is not a submodule of . Then there are distinct such that .

Proof. It suffices to show that is always closed under scalar multiplication of its elements by elements of . Let and . There is a nonzero element with such that , so ; hence, by Lemma 1.4, as required.

Theorem 3.2. Let be a module over a ring , and let such that is not a submodule of . Then one has the following. (i) is connected with .(ii)Some vertex of is adjacent to a vertex of . In particular, the subgraphs and of are not disjoint.(iii)If is connected, then is connected.

Proof. (i) Let . Then is adjacent to 0. Thus, is a path in of length two between any two distinct . Moreover, there exist nonadjacent by Lemma 3.1; thus, .(ii) By Lemma 3.1, there exist distinct such that . Then and are adjacent vertices in since . Finally, the “in particular” statement follows from Lemma 3.1.(iii) By part (i) above, it suffices to show that there is a path from to in for any and . By part (ii) above, there exist adjacent vertices and in and , respectively. Since is connected, there is a path from to in , and, since is connected, there is a path from to in . Then there is a path from to in since and are adjacent in . Thus, is connected.

Proposition 3.3. Let be a module over a ring , and let such that is not a submodule of . If the identity of the ring is a sum of zero divisors, then every element of the is the sum of at most -torsion elements.

Proof. Let and . We may assume that . Then there is a nonzero element such that , so with . Therefore, if and , then , so, for all , implies that , as needed.

Theorem 3.4. Let be a module over a ring , and let such that is not a submodule of . Then is connected if and only if is generated by its -torsion elements.

Proof. Let us first prove that the connectedness of the graph implies that the module is generated by its -torsion elements. Suppose that this is not true. Then there exists which does not have a representation of the form , where . Moreover, since . We show that there does not exist a path from 0 to in . If is a path in , are -torsion elements and may be represented as . This contradicts the assumption that is not a sum of -torsion elements. The reverse implication may be proved in a similar way as in [6, Theorem  3.2].

We give here with an interesting result linking the -torsion element graph to the total graph of a commutative ring .

Theorem 3.5. Let be a module over a ring , and let . If is connected, then is a connected graph. In particular, for every .

Proof. Note that, if and , then (see Proposition 3.3). Now suppose that is connected, and let . Let be a path from 0 to 1 in . Then ; hence, is a path from to . As all vertices may be connected via , is connected.

Theorem 3.6. Let be a module over a ring , and let such that is not a submodule of . If every element of is a sum of at most -torsion elements, then . If is the smallest such number, then .

Proof. We first show that, by assumption, for every nonzero element of . Assume that , where . Set for . Then is a path from 0 to of length in . Let and be distinct elements in . We show that . If is a path from 0 to and is a path from 0 to , then, from the previous discussion, the lengths of both paths are at most . Depending on the fact whether is even or odd, we obtain the paths or from to of length . Assume that is the smallest such number, and let be the shortest representation of the elements as a sum of -torsion elements. From the previous discussion, we have . Suppose that , and let be a path in . It means, a presentation of the element as a sum of -torsion elements (see the proof of Theorem 3.4), which is a contradiction. This completes the proof.

Corollary 3.7. Let be a module over a ring , and let such that is not an ideal of and . If , then . In particular, if is finite, then .

Proof. This follows from Proposition 3.3 and Theorem 3.6. Finally, if is a finite ring such that is not an ideal of , then by [3, Theorem  3.4], as required.

By Lemma 3.1, the following theorem may be proved in a similar way as in [6, Theorem  3.5].

Theorem 3.8. Let be a module over a ring , and let such that is not a submodule of . Then one has the following. (i)Either or .(ii) if and only if .(iii)If , then .(iv)If , then or .

Example 3.9. Let denote the ring of integers modulo 6. We define the mapping by Then and . Now one can easily show that is not a submodule of and . Clearly, is connected with . Moreover, since , we conclude that the subgraphs and of are not disjoint. Furthermore, is connected since is connected.

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