On the Cozero-divisor Graphs of Commutative Rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in   R     W R  , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and   W R  a bR  b aR . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs     R x   and    R x        such as connectivity, diameter, girth, clique numbers and pla-narity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.


Introduction
Let R be a commutative ring with non-zero identity and let   Z R be the set of zero-divisors of R. For an arbitrary subset A of R, we put .The zero-divisor graph of R, denoted by , is an undirected graph whose vertices are elements of   with two distinct vertices a and b are adjacent if and only if ab = 0.
The concept of zero-divisor graph of a commutative ring was introduced by Beck [1], but this work was mostly concerned with colorings of rings.The above definition first appeared in Anderson and Livingston [2], which contained several fundamental results concerning the graph .The zero-divisor graphs of commutative rings have been studied by several authors.For instance, the preservation and lack thereof of basic properties of under extensions to rings of polynomials and power series was studied by Axtell, Coykendall and Stickles in [3] and Lucas in [4].Also Axtell, Stickles and Warfel in [5], considered the zero-divisor graphs of direct products of commutative rings.

Let
be the set of all non-unit elements of R. For an arbitrary commutative ring R, the cozero-divisor graph of R, denoted by , was introduced in [6], which is a dual of zero-divisor graph "in some sense".The vertex-set of were studied in [6].In this paper, we study the cozero-divisor graphs of the rings of polynomials, power series and the direct product of two arbitrary commutative rings.Also, we look at the preservation of the diameter and girth of the cozero-divisor graphs in some extension rings.Our results "in some sense" are the dual of the main results of [3][4][5].
Throughout the paper, R is a commutative ring with non-zero identity.We denote the set of maximal ideals and the Jacobson radical of R by and

 
U R is the set of all unit elements of R. By a local ring, we mean a (not necessarily Noetherian) ring with a unique maximal ideal.
In a graph G, the distance between two distinct vertices a and b, denoted by G , is the length of the shortest path connecting a and b, if such a path exists; otherwise, we set  

 
The girth of G, denoted by g G , is the length of the shortest cycle in G, if G contains a cycle; otherwise, .Also, for two distinct vertices a and b in G, the notation means that a and b are adjacent.A graph G is said to be connected if there exists a path between any two distinct vertices, and it is complete if it is connected with diameter one.We use n  : , is called the clique number of G. Obviously (cf.see [7, p. 289]).For a positive integer r, an r-partite graph is one whose vertex-set can be partitioned into r subsets so that no edge has both ends in any one subset.A complete r-partite graph is one in which each vertex is joined to every vertex that is not in the same subset.The complete bipartite graph (2-partite graph) with subsets containing m and n vertices, respectively, is denoted by A graph is said to be planar if it can be drawn in the plane so that its edges intersect only at their ends.A subdivision of a graph is any graph that can be obtained from the original graph by replacing edges by paths.A remarkable simple characterization of the planar graphs was given by Kuratowski in 1930.Kuratowski's Theorem says that a graph is planar if and only if it contains no subdivision of 5 K or 3,3 K (cf.[8, p. 153]).Also, the valency of a vertex a is the number of edges of the graph G incident with a.

R x
In this section, we are going to study some basic properties of the cozero-divisor graph of the polynomial ring   R x .To this end, we first gather together the wellknown properties of the polynomial ring   R x , which are needed in this section.
Remarks 2.1 Let  Since the polynomials x and 1 + x are non-units, In the following theorem, we show that is always connected and its diameter is not exceeding three.
Theorem 2. 2 The graph Proof.Since   R x is a non-local ring, by [1, Theorem 2.5], it is enough to show that, for every non-zero element x m   .On the other hand, by parts (ii) and The following proposition states that the diameter of The following corollary is an immediate consequence of Theorem 2.2 and Proposition 2.3.
In the next two theorems, we investigate the girth of the graph . Also, by parts (ii) and (iii) of Remarks 2.1, x and 1 . Thus,   f x is adjacent to both distinct vertices t x and 1 t x  .Moreover, it is easy to see that t x is adjacent to 1 t x  .Therefore we have the cycle Proof.Consider the elements ,1 . Now, since the polynomials x and 1 x  do not divide the is the required cycle.
In the next theorem we study the clique number of is infinity and hence the chromatic number is infinity.Proof.Let be a positive integer and consider the subgraph of This means that , and so Proof.In view of the proof of Theorem 2.8, for all positive integers , the cozero-divisor graph n Recall that a graph on n vertices such that  of the vertices have valency one, all of which are adjacent only to the remaining vertex a, is called a star graph with center a.Also, a refinement of a graph H is a graph G such that the vertex-sets of G and H are the same and every edge in H is an edge in G. Now, we have the following result.
Proposition 2.10 If there exists a maximal ideal m of R with , then there is a refinement of a star graph in the structure of Proof.Suppose that is a maximal ideal of R.Then, for every element with have that .Also .Hence, a is adjacent to b.Therefore, is a refinement of a star graph with center a.Now, by Remarks 2.1 (v), is an in- refinement of a star graph.

     R x
We begin this section with some elementary remarks about the rings of power series which may be valuable in turn.These facts can be immediately gained from the elementary notes about power series.
In the following proposition, we study the connectivity and diameter of . In this regard, we have the following two cases:  Case 1. Assume that i , for some and consider an element b in .We will show that   and look for a contradiction.
We have that and b are adjacent.
Case 2. Suppose that i , for all .First assume that , for some .Hence, there exist maximal ideals m and such that i a m  , for some non-zero element g i in R which is a contradiction because the vertices a i and b are adjacent.On the other hand, We claim that   f x is adjacent to , where t is the least non-zero power of Therefore  is connected and also, by considering the above cases, it is routine to check that .
In the next lemma, we investigate the adjacency in in the case that R is a local ring.Lemma 3.3 Assume that R is a local ring with maximal ideal m.Let Then we have the following statements: is adjacent to all non-zero elements of   J R ; and, where   . Then 0 0 and 0 1 .Since a 0 and g 0 belong to m, we have that which is a contradiction.Also we have that , for some   The following result, which is one of our main results in this section, states that 4 is always connected and also .
Proof.Owing to Proposition 3.2, the result holds in the case that R is non-local.Assume that R is a local ring with maximal ideal m.In view of part (iii) of Remarks 3.1, be two non-zero elements in that are not adjacent.We have the following cases for consideration: and .Then by Lemma 3.3 (i), we have that f x x . If , for some i, j, then by part (ii) of Lemma 3.3, , for all non-zero elements in m .c m Also, if , i j a b  , for all , and are the least non-zero powers of x in , i j , t t   f x and   g x , respectively, with t t   , then by Lemma 3.3 (iii), one can easily check that     . Finally, we may assume that for some positive integer i,  , for all j.Thus, by parts (ii) and (iii) of Lemma 3.3, we have the path , where c is a non-zero element in and is the least non-zero power of x in   g x .Case 3. Without loss of generality, we may assume that 0 0 a  and 0 0 b  .So, if , for some j, then in view of parts (i) and (ii) of Lemma 3.3, we have the path , where c is a non-zero element in m.
Moreover, if j m  , for all j, then by Lemma 3.3, we have    , where is a non-zero element in m and t is the least non-zero power of Therefore, the cozero-divisor graph   is connected and in view of the above cases, one can easily check that . The following lemma is needed in the sequel.
, where . So, we have 0 and b 0 = 0. Thus a = 0 which is a contradiction.Hence In the next theorem, we show that .
In the next theorem, we compute the clique number of is also infinity.Proof.For every positive integer , it is enough to construct a complete subgraph of   with vertices.To this end, let n be an arbitrary positive integer and .Then, by Lemma 3.5, it is easy to see that the subgraph with vertex-set is infinity and this implies that is infinity.
We end this section with the following theorem.Theorem 3.8 The cozero-divisor graph Proof.In view of the proof of Theorem 3.7, 5 K is a subgraph of Thus, by Kuratowski's Theorem, is not planar.
Throughout this section, R 1 and R 2 are two commutative rings with non-zero identities.We will study the cozerodivisor graph of the direct product of R 1 and R 2 .Note that an element  belongs to W if and only if a b e begin this section with the following lemma.

 
, a b or , for some , then every element in R with i-th component is adjacent to all elements in R with i-th component .
and assume on the contrary that the vertices 1 and are not adjacent in .Without loss of generality, suppose that n , for some non-zero element .Therefore i i i and hence is not adjacent to , which is a contradiction.
. Thus The following corollary follows immediately from Lemma 4.1.
such that they are not adjacent in .Then a is not adjacent to a in and b is not adjacent to in .
In the next lemma, we establish some relations between the adjacency in the graph and adjacency in both graphs and .
Proof.Without loss of generality, suppose that is not planar.So, by Kuratowski's Theorem (cf.[8, p. 153]), it contains a subdivision of 5 K or 3,3 K .Now, by Lemma 4.3 (ii), one can conclude that Proposition 4. 5 In , we have the following inequalities: Remark 4.6 Suppose that and In the following theorem, we invoke the previous lemmas to show that   is a complete bipartite graph whenever and are fields.
Remark 4.6, every element in V 1 is adjacent to all elements of V 2 and vice versa.Also, it is easy to see that there is no adjacency between vertices in R , for any , and The following theorem is one of our main results in a U  R .this section.
by Remark 4.9, whenever b 0  In the following th he girth we consider   .In this situation, we first show that   is a bip  artite graph.To this end, set a and b in , a is adjacent to b if and only if is an ideal generated by the element c in R. Some basic results on the structure of this graph and the relations between two graphs

K
to denote the complete graph with n vertices.Moreover, we say that G is totally disconnected if no two vertices of G are adjacent.For a graph G, let   number of the graph G, i.e., the minimal number of colors which can be assigned to the vertices of G in such a way that any two adjacent vertices have different colors.A clique of a graph is any complete subgraph of the graph and the number of vertices in a largest clique of G, denoted by

Theorem 4 .
10 The cozero-divisor graph 0 b  .Then, in view of can obtain the path above cases, it is easy to see that

2 ,
 .Proof. 1) Without loss of generality, such that a is adjace suppose that nt to b.Now, by   a b W R   Lemma 4.3 (i) and Remark 4.9, we have the cycle        