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On Graphs Related to Comaximal Ideals of a Commutative Ring
We study the co maximal graph , the induced subgraph of whose vertex set is , and a retract of , where is a commutative ring. For a graph which contains a cycle, we show that the core of is a union of triangles and rectangles, while a vertex in is either an end vertex or a vertex in the core. For a nonlocal ring , we prove that both the chromatic number and clique number of are identical with the number of maximal ideals of . A graph is also introduced on the vertex set , and graph properties of are studied.
In 1988, Beck  introduced the concept of zero-divisor graph for a commutative ring. Since then a lot of work was done in this area of research. Several other graph structures were also defined on rings and semigroups. In 1995, Sharma and Bhatwadekar  introduced a graph on a commutative ring , whose vertices are elements of where two distinct vertices and are adjacent if and only if . Recently, Maimani et al. in  named this graph as the comaximal graph of and they noted that the subgraph induced on the subset is the key to the co-maximal graph. Many interesting results about the subgraph were obtained in Maimani et al.  and Wang , and their works show that the properties of the graph are quite similar to that of the modified zero-divisor graph by Anderson and Livingston . For example, both graphs are simple, connected, and with diameter less than or equal to three, and each has girth less than or equal to four if they contain a cycle. Because of this reason, in this paper we use to denote the graph of . In Section 2, we discover more properties shared by both zero-divisor graph and the subgraph of . In particular, It is shown that the core of is a union of triangles and rectangles, while a vertex in is either an end vertex or a vertex in the core. For any nonlocal ring , it is shown that the chromatic number of the graph is identical with the number of maximal ideals of . In Section 3, we introduce a new graph on the vertex set
This graph is in fact a retract of the graph and is thus simpler than the graph in general, but we will show that they share many common properties and invariants.
Jinnah and Mathew in  studied the problem of when a co-maximal graph is a split graph, and they determined all rings with the property. In Section 4, we give an alternative proof to their main [6, Theorem 2.3]. In the co-maximal graph , each unit of is adjacent to all vertices of the graph while an element of only connects to units of . Temporally, we say is in the center of the graph . Related to the co-maximal relation, there is the concept of rings with stable range one. Recall that a ring (which needs not be commutative) has one in its stable range, if for any with , there is an element such that is invertible. For example, the following classes of rings have one in their stable range: zero-dimensional commutative rings, von Neumann unit-regular rings, and semilocal rings. The concept comaximal graph gives an interesting graph interpretation of such rings. In fact, a commutative ring has one in its stable range if and only if for any pair of adjacent vertices in the co-maximal graph , the additive coset (and ) has at least one element in the center of the graph .
Throughout this paper, all rings are assumed to be commutative with identity. For a ring , let be the set of invertible elements of and the Jacobson radical of . Recall that a graph is called complete (resp., discrete) if every pair of vertices is adjacent (resp., no pair of vertices is adjacent). We denote a complete graph by and a complete (resp., discrete) graph with vertices by (resp., ). A subset of the vertex set of is called a clique if any two distinct vertices of are adjacent; the clique number of is the least upper bound of the size of the cliques. Similarly, we denote by the complete bipartite graph with two partitions of sizes , respectively. Recall that is called a spanning subgraph of a graph if and in implies in for all distinct vertices of , where means that and is adjacent to . Recall that a cycle in a graph is a path together with an additional edge (). For a simple graph and a nonempty subset of , there is the subgraph induced on : the vertex set is and the edge set is
2. The Subgraph
As noted by Maimani et al. in , the main part of the co-maximal graph is the subgraph induced on the vertex subset . In fact, we have the following facts. (O1) A vertex in is adjacent to every vertex of , while an element of only connects to units of . In fact, there is a sequential sum decomposition (O2) is empty if and only if is a local ring, that is, a commutative ring with a unique maximal ideal.
The authors of [3, 4] studied this subgraph and obtained many interesting results. In particular, it is proved that the graph is connected with diameter less than or equal to three [3, Theorem 3.1], that the girth of the graph is less than or equal to four [4, Corollary 3.8]. We include a detailed proof for the following fundamental property of graphs related to algebras.
Theorem 1 (see [3, Theorem 3.1]). The graph is connected with diameter less than or equal to three.
Proof. For any , set . Then for each , . For distinct , we claim(1) if and only if ,(2) if and only if .
Now for distinct , if , then there exists such that . Then clearly there is a path in and hence . If , then take any such that . We claim that and it will follow that . In fact, assume to the contrary that . Then we have
It follows that since . Then , a contradiction. This completes the proof.
Theorem 2. For any non-local ring , let . Then the following statements are equivalent.(1) is a bipartite graph.(2) is a complete bipartite graph.(3) has exactly two maximal ideals.(4), where each is a field.
Proof. : Assume that are the maximal ideals of . Then is a complete bipartite graph with two partitions and .
: Assume that is a complete bipartite graph with vertex partition . Then for any maximal ideal , is entirely contained in a single partition. Assume . If has a third maximal ideal , then for some . This is impossible since .
and : Clear.
: Clearly, there is no loss to assume .
If has only a finite number of maximal ideals, say , , set . Then , and each vertex in is adjacent to all verices in . This implies and hence , when has only a finite number of maximal ideals. If further is a bipartite graph, then ; that is, has exactly two maximal ideals.
In the following, assume that has infinitely many maximal ideals, and we proceed to prove . Assume that has a nontrivial idempotent . Then . If has no nontrivial idempotent element, then and hence . Then assume that both and are nonprimitive idempotents. By induction, for any integer , there exist nontrivial corner rings of such that . Set
Clearly is a clique in , and thus .
: This follows from the Chinese remainder theorem.
Note that the proof together with [4, Theorem 3.9(1)] actually gives an alternative proof to the fact that whenever is not a local ring.
Recall that a ring is called an exchange ring if the left module has the exchange property, see  and the included references for details. Recall that idempotents can be lifted modulo every ideal of an exchange ring . The class of exchange rings include artinian rings, semiperfect rings, and clean rings, the rings in which each element is a sum of an idempotent and a unit. Recall that for a ring with all idempotents central in , is clean if and only if is an exchange ring. For commutative clean rings , we have the following.
Corollary 3. For any commutative non-local exchange ring , let . Then is a bipartite graph if and only if is a complete bipartite graph, if and only if , where each is a local ring.
(O3) is totally disconnected if and only if it is an empty graph, which is true if and only if is a local ring.
Theorem 4 (see [3, 4]). For a ring , let . Then the following statements are equivalent.(1)A star graph is a spanning subgraph of , that is; has at least two vertices, and there exists a vertex in which is adjacent to every other vertex.(2) is a tree; that is, is nonempty, connected and contains no cycles.(3) is a star graph.(4) is isomorphic to for some field .
Corollary 5. For a finite simple graph with , assume for some ring . Then the following are equivalent:(1)A star graph is a spanning subgraph of .(2) is a tree; that is, contains no cycle.(3) for some prime number and some positive integer .
The works of [3, 4] show that the graph has many properties which the zero-divisor graph of a ring (or a semigroup) already has. Recall from [10, 11] that the core of a zero-divisor graph is always a union of triangles and squares, and a vertex in is either an end vertex or a vertex of the core. Recall that the core of a graph is by definition the subgraph induced on all vertices of cycles of . In the final part of this section, we will show that the graph has the same property.
Lemma 6. For any path in the graph , if is not a unit of , then the path is contained in a subgraph isomorphic to .
Proof. The given condition implies . Hence and in particular, . Furthermore, we have
Now assume . Then . If , then there is a subgraph in which contains the path , see Figure 1. Since , it follows that , and this completes the proof.
Lemma 7. If a vertex of is in a cycle of five vertices, then is in either a triangle or a rectangle.
Proof. Assume is a cycle in , and is not in any triangle, see Figure 2. We proceed to verify that is in a square.
In fact, if for some , then there is a square in . Thus in the following, we assume where . Then by Lemma 6, we can assume further
Note that is adjacent to , and is adjacent to in . If , then by we have . Then there is a rectangle in . Therefore we may assume .
Since , we can assume there is a path . If , then there is a rectangle . If , then and hence by there is a cycle . Then there is a rectangle and a triangle . This completes the verification.
Lemma 8. Let and assume that contains a cycle. Then for any path in the core of , is in a cycle with .
is a cycle in . We proceed to verify that is in a cycle with . Since , we have
Then . It follows that . Consider . If , then it follows that . In this case, there is a rectangle . In the following, assume , that is, , and let . Without loss of generality, assume and is a path from to . If further , then there is a rectangle . If , then there is a cycle . This completes the proof.
Theorem 9. For a ring , let and assume that contains a cycle. Then the following hold.(1)The core of is a union of triangles and rectangles, and the girth of is less than or equal to four.(2)Every vertex of is either an end vertex or a vertex of the core.
Proof. (1) The first statement follows from Lemmas 7 and 8.
(2) For the second statement, let be a path and assume to the contrary that is not in the core of . Since contains a cycle, by Theorem 1, we can assume that is a path, where both and are in the core. Clearly, . By Lemma 6, both and are units of . By assumption on , we can assume further
If , then we have , a contradiction. Then and , where . Thus we can replace by . In a similar way, replace by . Now in there is a path , where both and are in the core of . Certainly, there is a vertex in the core such that is adjacent to . Then a similar argument shows for some . Since , and the distance from to is at most three, we must have a triangle in the graph . Then implies , a contradiction. This completes the proof.
3. A Retract of
For simple graphs and , recall that if there exists a map such that for distinct , in implies and in . Such a map is called a graph homomorphism from to . If is a subgraph of , and there is a graph homomorphism , where the restriction of on is an identity, then is called a retract of , see Figure 3 for an example. If has no proper retract, then is called a core graph (e.g., is a core graph.). Note that the core of a graph needs not to be a core graph. Recall that the chromatic number of is the least positive integer such that there exists a graph homomorphism . It is the least number of colors needed for coloring the vertices of in such a way that no two adjacent vertices have a same color. The girth of a graph , denoted by , is the length of the minimal cycle in .
In this section, we introduce a new graph which is a retract of and we study this new graph.
Definition 10. For a ring and any , let . Construct a simple graph in the following and denote it as :
Clearly, this graph has less vertices and less edges than that of the graph in general, see Figure 3. More precisely, we have the following.
Proposition 11. (1) For a ring , the graph is a retract of .
(2) For any ideal of contained in , is a retract of .
Furthermore, we have the following.
(3) If the girth of is three, then so is the girth of .
(4) is connected with diameter less than or equal to three.
(5) and .
(6) and are homomorphically equivalent to the same unique core graph.
Proof. Clearly, is a surjective graph homomorphism. For any , fix a vertex in to obtain a subgraph of . Then the graph is isomorphic to the subgraph of . Thus is a retract of the graph . In a similar way, one checks . All the remaining results follow from . We check and in the following.
The result is clear by the definition of . In fact, if and has odd girth, then it is known that , since any cycle with odd length is a core graph.
(5) Since a composition of graph homomorphisms is still a graph homomorphism, clearly . On the other hand, since is isomorphic to a subgraph of , for any graph homomorphism , the restriction is also a graph homomorphism. This shows the second assertion of (5). The first equality holds by the definition of a retract of a graph. Furthermore, it can be proved that the equalities hold for any graph retract and any graph blowup. See  for details.
Corollary 12. (1) For any non-local ring , .
(2) is a complete bipartite graph if and only if has exactly two maximal ideals, which is true if and only if is a bipartite graph.
Example 13. Consider the ring . For this ring, , while . We draw the two graphs in Figure 3.
Recall a convenient construction from graph theory, the sequential sum of a sequence of graphs with for all . The vertex set of is and the edge set of is , where
We illustrate the construction in Figure 4 for the sequence of graphs , where is a triangle together with three end vertices adjacent to distinct vertices.
Theorem 14. Let be any commutative ring that is not a local ring, and let . Then the following numbers are identical:(1)the chromatic number ,(2)the clique number ,(3)the cardinal number of ,(4), (5).
Proof. First, recall from [2, Theorem 2.3] that
Then consider the following decomposition of the co-maximal graph into sequential sums of three subgraphs: where is a discrete subgraph while is a complete subgraph. Since , we have
By [4, Theorem 3.9(2)], . Then
By Proposition 11, we have , and the result follows from .
Now we study the interplay between the graph structure of and the algebraic property of . First we have the following
Proposition 15. For a ring , let . Then the following are equivalent:(1)A star graph is a spanning subgraph of .(2) is a star graph.(3), where is a field and is a local ring.
Proof. We only need to prove . Assume is adjacent to every other vertex in . Then . Assume . Then and . So we can assume at the start that is a nontrivial idempotent. Let . Then is a maximal ideal of and . Thus is a field. Now , and for any , we have . Thus is a local ring with the maximal ideal .
Note that for a finite local ring , if in which , then .
Corollary 16. For any commutative ring , if and only if , where each is a field.
Proof. If , then clearly . Conversely, assume . Then is a complete graph. By Proposition 15, we have , where is a field and is a local ring. If is not a field, then take any nonzero . Then , and is not adjacent to in , a contradiction. This completes the proof.
Proposition 17. For any commutative non-local ring , if and only if one of the following conditions holds. (1) is a prime ideal of .(2) has exactly two maximal ideals, and for any fields .
Corollary 18. For any commutative non-local ring , the graphs and have the same diameter if and only if for any fields but .
At the end of this section, we pose the following problem.
Question 1. Which rings have the property that is a generalized split graph? Which rings have the property that is a (generalized) split graph?
Recall from  that a simple graph is a generalized split graph if where the induced subgraph on (resp., on ) is a core graph (resp., a discrete graph). Note that is not a generalized split graph, while is a split graph.
4. A Ring Whose Comaximal Graph Is a Split Graph
Throughout this section, assume that is a split graph, that is, is simple and connected with , where and the induced subgraph on (resp., on ) is a complete (resp., discrete) graph, see  and the included references there. For the split graph , we always assume that is a maximal such independent subset. Under this assumption, a complete graph is a split graph with .
Lemma 19. For a commutative ring , let be the co-maximal graph . If is a split graph with , then we have the following.(1)For any proper ideal of , .(2) holds for distinct maximal ideals of .(3)If is isomorphic to neither nor , then . Also, if and only if or .
Proof. (1) Clear.
(2) Assume to the contrary that there exists some nonzero . Since , we have such that . Clearly, and hence we can assume . Then it follows from Lemma 19 that , a contradiction.
(3) If is a local ring, then either or has at least two units by assumption. The result holds in this case. In the following, we assume that is non-local. If there is a maximal ideal with , then by the proof of [6, Theorem 2.1], for some field . In this case, and, if and only if . If no maximal ideal is contained in , then each maximal ideal has exactly one vertex in by . By (2), we have .
The following is a result of .
Lemma 20 (see [6, Theorem 2.1]). For a commutative ring which is non-local, if the co-maximal graph is a split graph and for any field , then the characteristic of is two, has exactly three maximal ideal , and for all , where each is idempotent and for every invertible element of , .
We now give an alternative proof to the main result of .
Theorem 21 (see [6, Theorem 2.3]). For a commutative ring , the co-maximal graph is a split graph if and only if is one of the following: a local ring, , for some field .
Proof. We only need to prove the necessary part. Assume that is not a local ring and for any field . Let . Assume further that the co-maximal graph is a split graph with . By Lemma 20, has exactly three maximal ideals () such that , where and hold for every . Also, has characteristic 2. For any , assume with . By Lemma 19, we have and hence . It follows that . Note that since , and it follows that . Then implies for any rearrangement of . Therefore . This shows that is a complete set of orthogonal idempotent elements of .
Let . Then , where each is a local ring with a unique maximal ideal , since has exactly three maximal ideals. For any , we have
Since for all , it follows that . Now apply the fact ; one derives . Hence and each is a field. Thus is a direct product of three fields.
Since , it follows that
Then in the decomposition , we deduce from Lemma 19(1) that
Now we claim that each is isomorphic to and hence . In fact, assume . Then
If , then there exists a nonzero element . Then both and are in the independent subset , contradicting . This completes the proof.
Corollary 22. A finite split graph is a co-maximal graph of a non-local commutative ring if and only if is one of the following: (1) the sequential sum for some prime number and (2) the sequential sum in Figure 4, where is a triangle together with three end vertices adjacent to distinct vertices.
The co-maximal graph of a field is certainly a complete graph. For a finite local ring which is not a field, assume . Then the co-maximal graph of is , where and is a discrete graph with vertices.
By Theorem 21 and the results of [3, 4], it is natural to ask the following question: for what rings is a split graph, where the vertex set of is ? Note that for distinct maximal ideals (if exist) and , implies . Then a careful check to the proofs of Lemma 19, Lemma 20, and Theorem 21 shows the following.
Theorem 23. For any ring , the following statements are equivalent.(1)The co-maximal graph of is a split graph.(2)The subgraph is either empty or a split graph.(3) is one of the following: a local ring, , for some field .
Note that a split graph is isomorphic to for some finite ring if and only if either is a star graph for some prime number or is the triangle together with three end vertices adjacent to distinct vertices (see the graph in Figure 4).
This research is supported by the National Natural Science Foundation of China (Grant no. 11271250).
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