1. Introduction
Let R be a commutative ring with non-zero identity and let
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
is
and for two distinct vertices a and b in
, a is adjacent to b if and only if
and
, where cR 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
and
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 [3-5].
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
, respectively. Also,
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
, is the length of the shortest path connecting a and b, if such a path exists; otherwise, we set
. The diameter of a graph G is

The girth of G, denoted by
, 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
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
denote the chromatic 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
, 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
or
(cf. [8, p. 153]). Also, the valency of a vertex a is the number of edges of the graph G incident with a.
2. Cozero-Divisor Graph of 
In this section, we are going to study some basic properties of the cozero-divisor graph of the polynomial ring
. To this end, we first gather together the wellknown properties of the polynomial ring
, which are needed in this section.
Remarks 2.1 Let
be an arbitrary element in
. Then we have the following statements:
•
is a unit in
if and only if a0 is a unit and the coefficients
are nilpotent elements of R.
•
is nilpotent if and only if the coefficients
are nilpotent.
•
, where
is the nilradical of
.
• Since the polynomials x and 1 + x are non-units,
is a non-local ring.
• By part (i), it is easy to see that
is an induced subgraph of
.
In the following theorem, we show that
is always connected and its diameter is not exceeding three.
Theorem 2.2 The graph
is connected and
.
Proof. Since
is a non-local ring, by [1, Theorem 2.5], it is enough to show that, for every non-zero element
, there exist
and
such that
. Now, assume that
is a non-zero polynomial in
of degree t. Since x is a non-unit element in
, there exists a maximal ideal m of
such that
. So
. On the other hand, by parts (ii) and (iii) of Remarks 2.1,
. Also
. Hence the graph
is connected and
.
The following proposition states that the diameter of
is never one.
Proposition 2.3 The graph
is never complete.
Proof. Clearly
and
. The claim now follows from [1, Theorem 2.1].
The following corollary is an immediate consequence of Theorem 2.2 and Proposition 2.3.
Corollary 2.4
or 3.
Proposition 2.5 Suppose that
. Then
. In particular, if R is reduced, then
.
Proof. In view of [1, Corollary 2.4],
. Now, by Proposition 2.3, one can conclude that
. Also, if R is reduced, then by Remarks 2.1 (ii),
and so, by Remarks 2.1 (iii),
.
In the next two theorems, we investigate the girth of the graph
.
Theorem 2.6 Suppose that R is a non-reduced ring. Then every element of
is in a cycle of length three.
Proof. Assume that
is a non-zero element in
of degree n and consider the elements
and
in
, where
. Then, by Remarks 2.1 (iv), there exist maximal ideals m1 and m2 of
such that
and
. Since t > n,
and
. Also, by parts (ii) and (iii) of Remarks 2.1,
and
do not belong to
. So,
and
. Thus,
is adjacent to both distinct vertices
and
. Moreover, it is easy to see that
is adjacent to
. Therefore we have the cycle

Theorem 2.7
.
Proof. Consider the elements
and
in
. So there exist two maximal ideals m1 and m2 in
such that
and
. Hence the vertex
is adjacent to
. Also, clearly
and
. Now, since the polynomials x and
do not divide the polynomial
, we have that
and
. Hence

is the required cycle.
In the next theorem we study the clique number of
.
Theorem 2.8 In the graph
,
is infinity and hence the chromatic number
is infinity.
Proof. Let
be a positive integer and consider the subgraph
of
with vertex-set
. Now, for every two distinct polynomials
and
with i < j, clearly we have that

Also, since
, we have that
. This means that
, and so
does not divide the polynomial
. Thus
. Hence,
is a complete subgraph of
which is isomorphic to
. So
is infinity. This implies that
is infinity.
Theorem 2.9 The cozero-divisor graph
is not planar.
Proof. In view of the proof of Theorem 2.8, for all positive integers
, the cozero-divisor graph
has a complete subgraph isomorphic to
. In particular, for
, the graph
is a subgraph of
. So, by Kuratowski’s Theorem (cf. [8, p. 153]),
is not planar.
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
, we 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 induced subgraph of
. So
contains a refinement of a star graph.
3. Cozero-Divisor Graph of 
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.
• Remarks 3.1
•
is a unit in
if and only if
is a unit in R.
•
belongs to the Jacobson radical of
if and only if
belongs to the Jacobson radical of R.
•
is a local ring if and only if
is local.
• The cozero-divisor graph
is an induced subgraph of
, but
is not a subgraph of
, since 1 + x is a vertex of
but it is not in the vertex-set of
.
In the following proposition, we study the connectivity and diameter of
, whenever R is non-local.
Proposition 3.2 Let R be a non-local ring. Then the cozero-divisor graph
is connected and
.
Proof. Suppose that
is a non-zero element in
. By [6, Theorem 2.5], it is enough to show that
is adjacent to some element in
. In this regard, we have the following two cases:
Case 1. Assume that
, for some
and consider an element b in
. We will show that
is adjacent to b. Clearly, by Remarks 3.1 (i)(ii),
. Now, assume in contrary that
and look for a contradiction.
We have that
, for some
in
. Since, by Remarks 3.1 (ii),
, we have that
which is impossible. Also, if
, then
, for some non-zero element gi in R, which is impossible. Therefore
and b are adjacent.
Case 2. Suppose that
, for all
. First assume that
, for some
. Hence, there exist maximal ideals m and
such that
. By considering an element b in
, one can conclude that ai is adjacent to b. Now if
, then
, for some non-zero element gi in R which is a contradiction because the vertices ai and b are adjacent. On the other hand,
. Thus the vertices
and b are adjacent.
Now, let
, for all
. Choose
. Hence
. We claim that
is adjacent to
, where t is the least non-zero power of
in the polynomial
. Clearly,
. Now, if
for some
in
, then

which belongs to
and this is impossible. Hence, we have that
is adjacent to
.
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
be a non-zero element in
. Then we have the following statements:
• if
, then
is adjacent to
;
• if
and
, for some
, then
is adjacent to all non-zero elements of
; and• if
and
, for all
, then
is adjacent to
, where
is the least non-zero power of
in
.
Proof. 1) Assume on the contrary that
, where
. Then
and
. Since a0 and g0 belong to m, we have that
which is a contradiction. Also we have that
. Thus
is adjacent to x.
2) Let b be a non-zero element in m. Then if
, we conclude that
which is impossible. So
. Now, since
. Therefore,
is adjacent to
.
3) Clearly, since
is the least non-zero power of
in
,
. Moreover, if
, for some
in
, then

This means that
which is impossible. Hence
. So
is adjacent to
.
The following result, which is one of our main results in this section, states that
is connected and the diameter of
is not exceeding four.
Theorem 3.4 The cozero-divisor graph 
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,
is also a local ring. Now, let
and
be two non-zero elements in
that are not adjacent. We have the following cases for consideration:
Case 1.
and
. Then by Lemma 3.3 (i), we have that
.
Case 2.
. If
, for some i, j, then by part (ii) of Lemma 3.3,
, for all non-zero elements
in
.
Also, if
, for all
, and
are the least non-zero powers of x in
and
, respectively, with
, then by Lemma 3.3 (iii), one can easily check that
.
Finally, we may assume that for some positive integer i,
and
, 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
.
Case 3. Without loss of generality, we may assume that
and
. So, if
, for some j, then in view of parts (i) and (ii) of Lemma 3.3, we have the path
, where
is a non-zero element in m.
Moreover, if
, 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
in
.
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.
Lemma 3.5 Let
and let i and j be positive integers such that
. Then the vertices
and
are adjacent in
.
Proof. Suppose to the contrary that
, where
is a non-zero polynomial in
. So, we have
and b0 = 0. Thus a = 0 which is a contradiction. Hence
. Also, clearly
. So the vertices
and
are adjacent in the cozero-divisor graph
.
In the next theorem, we show that
.
Theorem 3.6 The cozero-divisor graph
has girth three.
Proof. Let
. Consider the elements x,
and
in
. Clearly,
and
. Also, since
,
and
don’t belong to
. Hence, we have the following path

Now, in view of Lemma 3.5, one can conclude that
and
are adjacent. Therefore, we have the cycle
. Hence,
.
In the next theorem, we compute the clique number of
.
Theorem 3.7 In the graph
,
is infinity and hence
is also infinity.
Proof. For every positive integer
, it is enough to construct a complete subgraph of
with
vertices. To this end, let
be an arbitrary positive integer and
. Then, by Lemma 3.5, it is easy to see that the subgraph with vertex-set
is a complete subgraph of
which is isomorphic to
. So
is infinity and this implies that
is infinity.
We end this section with the following theorem.
Theorem 3.8 The cozero-divisor graph
is not planar.
Proof. In view of the proof of Theorem 3.7,
is a subgraph of
. Thus, by Kuratowski’s Theorem,
is not planar.
4. Cozero-Divisor Graph of 
Throughout this section, R1 and R2 are two commutative rings with non-zero identities. We will study the cozerodivisor graph of the direct product of R1 and R2. Note that an element
belongs to
if and only if
or
. We begin this section with the following lemma.
Lemma 4.1 Suppose that
is a direct product of finite commutative rings. If
is adjacent to
in
, for some
, then every element in R with i-th component
is adjacent to all elements in R with i-th component
.
Proof. Suppose that
is adjacent to
in
and assume on the contrary that the vertices
and
are not adjacent in
. Without loss of generality, suppose that
. Thus
, for some non-zero element
. Therefore
and hence
is not adjacent to
, which is a contradiction.
The following corollary follows immediately from Lemma 4.1.
Corollary 4.2 Suppose that
,
such that they are not adjacent in
. Then a is not adjacent to
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
.
• Lemma 4.3
• Let
and
. Then
is adjacent to
in
if and only if b is adjacent to
in
.
• Let
and
. Then
is adjacent to
in
if and only if
is adjacent to
in
.
Proof. 1) Suppose that
is adjacent to
. Note that if at least one of the elements
or
is zero or unit, then
is not adjacent to
in
. Thus we can suppose that
. Now, if
is not adjacent to
, then
or
. So without loss of generality, we may assume that
for some non-zero element
. Hence
. This means that
and
are not adjacent in
which is impossible. Therefore
and
are adjacent in
. Conversely, if
is adjacent to
, then by Lemma 4.1, we have that
is adjacent to
.
2) The proof is similar to part 1).
The following propositions follow directly from Lemma 4.3.
Proposition 4.4 Assume that either
or
is not planar. Then
is not planar.
Proof. Without loss of generality, suppose that
is not planar. So, by Kuratowski’s Theorem (cf. [8, p. 153]), it contains a subdivision of
or
. Now, by Lemma 4.3 (ii), one can conclude that
is not planar.
Proposition 4.5 In
, we have the following inequalities:

;
•
.
Remark 4.6 Suppose that
and
. Then
is adjacent to
.
In the following theorem, we invoke the previous lemmas to show that
is a complete bipartite graph whenever
and
are fields.
Theorem 4.7 Assume that
and
are fields. Then
is a complete bipartite graph.
Proof. Put
and
. Clearly
. By Remark 4.6, every element in V1 is adjacent to all elements of V2 and vice versa. Also, it is easy to see that there is no adjacency between vertices in V1 (or V2). So
is a complete bipartite graph.
Corollary 4.8 Let
be an arbitrary field. Then
and
are star graphs.
Remark 4.9 It is easy to see that
is adjacent to
in
, for any
,
and
. Similarly,
is adjacent to
, for any
,
and
.
The following theorem is one of our main results in this section.
Theorem 4.10 The cozero-divisor graph
is connected and
.
Proof. Suppose that
and
are arbitrary elements in
. We have the following cases for consideration:
Case 1.
. If
, then consider the path
. If
and
, then, by Remark 4.9, we have that
. Now, suppose that
and
. Then, in view of Remarks 4.6 and 4.9, one can obtain the path
in
. The similar result holds in the case that
and
.
Case 2.
and
. If
, then, by Remark 4.9, whenever
, we have that
. Otherwise,
. Since
and
, we have
. Now, if
, then
. But
and this implies that
which is not true. Hence
. Also, since
, it is easy to see that
. Therefore, we have the path
. Also, if
, then by Remarks 4.6 and 4.9, we can consider the path
. The similar result holds if
and
.
Case 3.
. Then we have that
, and we can apply Case 1 on the second component of ordered pairs.
Now, in view of the above cases, it is easy to see that
.
In the next proposition, we provide a characterization of the complete cozero-divisor graph
.
Proposition 4.11 The graph
is complete if and only if
is isomorphic to
.
Proof. If
, then
is not adjacent to
, for some
. Similarly, if
, then
is not complete. So, if
is complete, then
. Also, clearly the graph
is complete.
Corollary 4.12 If
, then
or 3.
In the following theorem, we study the girth of
. Note that we consider
to be totally disconnected.
• Theorem 4.13
• If at least one of the cozero-divisor graph
or
is not totally disconnected, then
.
• If
and
, then
.
• If
and R2 is a field, then
.
• If
and R2 is not a field, then
or
.
Proof. 1) Without loss of generality, suppose that
such that a is adjacent to b. Now, by Lemma 4.3 (i) and Remark 4.9, we have the cycle
in
.
2) Let
and
such that a and b are not identity. Now, consider the cycle
in
.
3) By Corollary 4.8,
is a star graph and so
.
4) First, assume that
. Let
and
. Now, by Remarks 4.6 and 4.9, we have the cycle
in
. In the case that
, if
is not totally disconnected, then by part 1),
. So, assume that there is no adjacency in
. In this situation, we first show that
is a bipartite graph. To this end, set
and
. Clearly,
. Also, by Lemma 4.3, no two vertices in
(or
) are adjacent. So
is a bipartite graph, and thus
or
. Now, by Remark 4.9,
is adjacent to all vertices
in
, where
, and also
is adjacent to all vertices
in
, where
. Hence, if there exist an element
and
such that
is adjacent to
in
, then
. Otherwise, the girth of the cozero-divisor graph
is infinity.
The following example presents a ring
with
which satisfies parts 1) and 4) of Theorem 4.13. This shows that all cases in the proof of the last part of Theorem 4.13, can occur.
Example 4.14 Let
and
. Then
and by Proposition 4.11,
is complete. Hence, by Theorem 4.13 (i), we have that
.
5. Acknowledgements
The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions.
NOTES