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.