Abstract
Let R be a commutative ring with unity and let M be a unitary module. In this paper, we derive some completeness conditions on the zero divisor graphs of modules over commutative rings. It is shown that the weak zero divisor graph of a simple module is complete if and only if R is a field. We investigate the zero divisor graphs in finitely generated modules. We find the diameter, the girth, the clique number and the vertex degrees of the zero-divisor graphs of the rings of integer modulo n as modules.
1. Introduction
A simple graph G consists of a vertex set and an edge set E(G) of unordered pairs of distinct vertices. The cardinality of V(G) is called the order of G and the cardinality of E(G) is its size. A graph G is connected if and only if there exists a path between every pair of vertices u and v. A graph on n vertices such that every pair of distinct vertices is joined by an edge is called a complete graph, denoted by Kn. A complete subgraph of G of largest order is called a maximal clique of G and its order is called the clique number of G, denoted by cl(G). The number of edges incident on a vertex v is called the degree of v and is denoted by dv or d(v). A vertex of degree 1 is called a pendent vertex. In a connected graph G, the distance between two vertices u and v is the length of the shortest path between u and v. The diameter of a graph G is defined as where d(u, v) denotes the distance between vertices u and v of G. For more definitions and terminology of graph theory, we refer to [Citation9].
Throughout, R shall denote a commutative ring with unity Let Z(R) be the set of zero-divisors of R. The concept of the zero-divisor graph of a commutative ring was first introduced by Beck [Citation4]. The zero-divisor graph associated to a ring R has its vertices as elements of and two vertices are adjacent if and only if xy = 0.
We denote a unitary module by M, unless otherwise stated. For an module M and the set is clearly an ideal of R and an annihilator of the factor module M/Rx. The annihilator of M denoted by ann(M) is The concept of the zero-divisor graph has been extended to modules over rings, see for instance, [Citation5, Citation10, Citation11]. Further, Ghalandarzadeh and Rad [Citation6] extended the notion of the zero-divisor graph to the torsion graph associated with a module M over a ring R, whose vertices are the non-zero torsion elements of M such that two distinct vertices a and b are adjacent if and only if The idea was extended to other graph structures, like the zero-divisor graphs of idealizations with respect to the prime modules [Citation2], the L-total graph of an L-module [Citation1], etc, to mention a few.
For any set X, let denote the cardinality of X and denote the set of the non-zero elements of X. We denote an empty set by and the complement of X shall be denoted by Xc. We denote the ring of integers by the ring of integer modulo n by and the finite field with q elements by For more definitions and terminology of module and ring theory, we refer to [Citation3, Citation7].
The rest of the paper is organized as follows. In Section 2, we include some completeness conditions of the zero-divisor graph of the unitary modules. For instance, it is shown that the zero divisor graph of is complete for every simple module M. In Section 3, we investigate some graph parameters of the zero-divisor graphs of the modules like the diameter, the girth, the clique number and the vertex degrees.
2. Graphs associated with modules over commutative rings
Throughout, we treat M as a unitary module. Let We define the annihilator of N by For we denote the annihilator of the factor module M/Rm by Thus, Let z be an element in M. The following definition is due to Behboodi [Citation5]. An element is a
weak zero divisor, if either z = 0 or for some with
zero divisor, if either z = 0 or and for some with
strong zero divisor, if either z = 0 or and for some with
For any module M, we write Z(M) and respectively, for the set of non-zero weak zero divisors, non-zero zero divisors and non-zero strong zero divisors. Clearly, and all of these sets coincide with the set of zero divisors of R when M = R. Behboodi [Citation5] associated three simple graphs, denoted by and called the weak zero-divisor graph, zero-divisor graph and strong zero-divisor graph, to an module M with vertex sets defined as Z(M) and respectively. Two distinct vertices and being adjacent if and only if From the definition, clearly as induced subgraphs.
Behboodi [Citation5] showed that for any module M, either or and also, is always connected with diameter at most 3. Moreover, if is not a tree, then the girth of is at most 4. Further, characterized the modules M for which and showed that such a property is only enjoyed by the multiplication modules. Whenever, we shall write with vertex set Behboodi showed that the weak zero-divisor graph of a module M is finite if and only if either M is finite or prime multiplication-like module.
Example 2.1.
Let and Then M consists of 12 elements as an module. As M is a multiplication-like module, we have Also, we have Further, it can be verified that Now, let Then For any we have The zero divisor graphs of M and are given in .
If X is a subset of a module M over a ring R, then the intersection of all submodules of M containing X is called the submodule generated by X (or spanned by X). If X is finite, and X generates the module M, then M is said to be finitely generated. If then X clearly generates the zero module. If X consists of a single element, say, then the submodule generated by X is called the cyclic (sub)module generated by a. Finally, if is a family of submodules of M, then the submodule generated by is called the sum of the modules Mi. If the index set I is finite, the sum of is denoted by A non zero module M is said to be simple if it has no submodules other than (0) and M.
The following theorem provides a condition for the adjacency of two distinct vertices in the zero divisor graph of a finitely generated module.
Theorem 2.2.
Let be a sequence of finitely generated simple modules and let . Then if and only if xR and yR are disjoint modules.
Proof.
Let Assume to the contrary and let Then, the submodule generated by z is given as So there exist subsets and of such that and Therefore, we can write In this notation, we have
Thus, we have Since we have This implies that
Now, since each is simple and for all we conclude that and are coprime. Therefore, we can write (2.1) (2.1)
This implies that for every there exists such that Therefore, and so Mq = Mp. Finally, So there exists such that Mk = Ms. As in EquationEquation (2.1)(2.1) (2.1) , there exists such that Thus, This implies that z = 0, which is contradicts the hypothesis. On the other hand, since we conclude that which implies that □
The following lemma will be used in the sequel.
Lemma 2.3.
[Proposition 5.3.4, [Citation8]] An module M is simple if and only if for some maximal ideal in R.
An module M is said to be decomposable if there exist two non-zero submodules M1 and M2 such that and indecomposable if it is not a direct sum of two non-zero submodules. The following theorem shows that the zero divisor graph of a simple R-module is complete.
Theorem 2.4.
If M is a simple module, then is complete.
Proof.
Let M be a simple module, and let By definition, for every Therefore, for each we have Similarly, Now, for each we have By Lemma 2.3, we see that is a maximal ideal of R, which is contained in Now, if we are done. Otherwise, which gives Therefore, there exists such that and Thus, which implies that Therefore a contradiction. Thus we have and so is complete. □
Let and (copies of ). Then if some zi = 0 and some for some and Thus, the strong zero divisor graph of M is empty and that
As seen above, is complete when is considered as a module. However, the same does not hold true in general for all non-simple modules M when the ring R is chosen arbitrarily. The following theorem restricts the choice for the ring R for a module M to have a complete zero divisor graph.
Theorem 2.5.
Let M be an module which is not simple. Then is complete if and only if R is a field.
Proof.
As M is not simple, there exists an submodule such that Let for some Then This implies that which is a contradiction. Therefore, Thus, for all we have Conversely. assume that for all Let N be a proper ideal of R. Consider and let where and Choose As we have Because for every we have rN = 0 for every This also implies that and hence □
Let M be a module. Let z be a non-zero weak zero divisor in M. Then for some It is trivial to see that for all Also, if and only if M is a simple module. Since every finite module M is a finite abelian group, so we have the following proposition.
Proposition 2.6.
A vertex in a weak zero divisor graph of a finite module M represents an essential ideal if and only if M is a non-simple finite group.
Let ( copies of ) be a module. Then it is easy to see that each non-zero element of M is a weak zero divisor and that for all we have Therefore, Now, the submodules generated by the non-zero weak zero divisors of M are the lines with integral coordinates in the hyperplane intersecting at the origin only. It follows that for every non-zero weak zero divisor m of M the ideal is not an essential ideal. This shows that Proposition 2.6 is not true for infinite modules.
Theorem 2.7.
Let R be an integral domain and M an module. If there exists an element such that , then is complete.
Proof.
Choose and Let Then for some This gives which implies that since R is a domain. This further implies that z = 0 because and Therefore, for each and we have Further, let and choose Then for some This gives and so Therefore, is complete. □
Definition 2.8.
Let R be a ring and M be an module. If for every non-zero submodule N of M and an ideal A of R with NA = 0 implies MA = 0, we say that M is a prime module. This is equivalent to saying that for every non-zero submodule N of M. It is immediate that is a prime ideal, and it is called the affiliated prime of M. Also, if each submodule of M is of the form AM for some ideal A of R, then we say that M is a multiplication module. Moreover, if a multiplication module M satisfies for every non-zero submodule N of M, then M is called a multiplication-like module.
Theorem 2.9.
Let M be a multiplication module over a ring R. Then the zero divisor graph of M is empty if and only if M is a prime multiplication-like module.
Proof.
Since every multiplication module is a multiplication-like module, therefore it suffices to prove the result for multiplication-like modules. Assume that M is not a prime multiplication-like module. We will show that is non-empty. As M is not a prime multiplication-like module, we have is not a prime ideal. Thus, there exist ideals and which properly contain and satisfy and Thus, we can find and such that and Then, we have Therefore,
On the other hand, if M is a prime multiplication-like module, then for every non-zero that is, for each non-zero we have Therefore, □
3. Graph parameters of zero divisor graphs of modules
In the following theorem, we compute the clique number of the zero divisor graph of a multiplication module. Noting that the weak zero divisor graph, the zero divisor graph and the strong zero divisor graph all coincide in case of multiplication-like modules, we write to denote the zero divisor graph of such modules.
Theorem 3.1.
Let M be an module, where and for and a prime p. Then the clique number of is equal to or according as t is even or odd.
Proof.
It follows immediately that every vertex of is of the form rp for some We divide the vertex set of into disjoint subsets where It is not difficult to see that the cardinality of as a subset of is equal to Let and be two vertices of Then if and only if Thus, for all we have for all integers Now, assume that t is even. Then for all and Also, when t is odd, no two vertices are adjacent inside and every vertex of is adjacent to every vertex of Therefore, it follows that when t is even and is equal to when t is odd. □
The girth of a graph G is defined as the length (or order) of the smallest cycle contained in G, and is denoted by gr(G). If G has no cycle, then The following theorem characterizes the diameter, the smallest () and the largest () vertex degree and the girth of the zero divisor graph of a module M.
Theorem 3.2.
Let p be a prime integer and . Then for an module M, where and , the following statements hold, unless
and
if and only if t = 4, 8, 9, otherwise
Proof.
As in the proof of Theorem 3.1, we define Then gives a partition of the vertex set of and Now, two elements and of M satisfy if and only if Therefore, it follows instantly that every vertex is adjacent to every vertex contained in Thus, Further, let and choose Then if and only if Thus, it follows that and that This proves (1) and (2).
(3) From the previous paragraph, we see that for all for all vertices of Also, if then only if Thus, the set of elements in form the center of Now, assume that is a tree. Then has either one or two centres. Therefore, must be either 1 or 2, so that 2 or 3. Now, either or form a triangle in for all and Therefore, the result follows. □
Corollary 3.3.
if and only if is a star graph, where , is considered a module.
Let where p and q are distinct primes, be a module. Then, it can be easily verified that if if and if Thus, is complete bipartite. While one expects that if are distinct primes, and then is complete partite, but this is not a case. However, contains the so expected partite graph as a subgraph as can be seen in in which, = and so that thus containing six different vertex degrees. However, a complete partite graph can possess at most t distinct vertex degrees. Therefore, is not complete partite, but we see that the subsets and of the vertex set of induce a complete partite subgraph.
If p, q, r are distinct primes and be a module, then it is always possible to partition the vertex of into six disjoint sets, say, where are defined in the following way. Let denote the arbitrary non-zero elements in and respectively. Then and Let Then it is an easy exercise to verify that and Thus, it can be easily seen that for is an element of the ordered set Moreover, the clique number of is 3 and the sets and induce a complete partite subgraph. In fact, this can be generalized to the following theorem.
Theorem 3.4.
Let M be an module, where and then
where i runs over the indices of in which are equal to 0.
The set of vertices induces a complete partite subgraph in
Theorem 3.5.
Let M and N be two modules such that the sum of their annihilators equals R. Then the following statements hold.
If then
If and then
If and then where denotes the number of elements η in cliques of and respectively, whose square is 0.
Proof.
For any module M, let and denote the set of weak zero divisors and non-zero weak zero divisors of M. Let M and N be two modules, and and
Assume that Then for each and we have and Thus,
As we have Let be an induced maximal clique in Then, for each there exists some such that Now, for each we have and for all and we have Thus, the vertices of the form contribute 1 to the clique number and the fact that we conclude that
Let and Let and be the induced maximal complete subgraphs of and Then, for each and we can find and which satisfy and Also, for every and there exist no and for which holds true. Even if m1 (or equivalently ) is chosen from M, then a similar statement holds true if (or equivalently ). A similar argument is valid for if chosen from N. Thus, such vertices do not contribute to the clique number. Now, for all and we have but however such a vertex, say contributes to the clique if and only if and This argument adds each vertex and which satisfy and to the clique. Therefore, the clique number of is equal to where η1 and η2 are the number of vertices and respectively, which satisfy and □
Theorem 3.6.
Let R be a finite integral domain and M be an module which is not simple. Then
Proof.
The proof follows by Theorem 2.5. □
Theorem 3.7.
Let be a module, where are distinct primes. Then the clique number of is equal to t, if . In case , then the clique number of is
Proof.
Let Define and choose Then for all Therefore, contains a clique of order t. Moreover, if then for all Hence, Now, let Then and we have Consider the submodules then we have for all and where Moreover, let be a submodule of M, where Then k is of the form where some ci < bi. Without loss of generality, let and let then we get where is an element of Therefore, the clique number is equal to □
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Acknowledgments
We are grateful to the anonymous referee for his useful suggestions.
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References
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