On Graphs Related to Comaximal Ideals of a Commutative Ring

Abstract

We study the co maximal graph Ω(𝑅), the induced subgraph Γ(𝑅) of Ω(𝑅) whose vertex set is 𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)), and a retract Γ_𝑟(𝑅) of Γ(𝑅), where 𝑅 is a commutative ring. For a graph Γ(𝑅) which contains a cycle, we show that the core of Γ(𝑅) is a union of triangles and rectangles, while a vertex in Γ(𝑅) is either an end vertex or a vertex in the core. For a nonlocal ring 𝑅, we prove that both the chromatic number and clique number of Γ(𝑅) are identical with the number of maximal ideals of 𝑅. A graph Γ_𝑟(𝑅) is also introduced on the vertex set {𝑅𝑥∣𝑥∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅))}, and graph properties of Γ_𝑟(𝑅) are studied.

1. Introduction

In 1988, Beck [1] 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 [2] introduced a graph Ω(𝑅) on a commutative ring 𝑅, whose vertices are elements of 𝑅 where two distinct vertices 𝑥 and 𝑦 are adjacent if and only if 𝑅𝑥+𝑅𝑦=𝑅. Recently, Maimani et al. in [3] named this graph Ω(𝑅) as the comaximal graph of 𝑅 and they noted that the subgraph Γ(𝑅) induced on the subset 𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)) is the key to the co-maximal graph. Many interesting results about the subgraph were obtained in Maimani et al. [3] and Wang [4], and their works show that the properties of the graph Γ(𝑅) are quite similar to that of the modified zero-divisor graph by Anderson and Livingston [5]. 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 Γ(𝑅) to denote the graph Γ₂(𝑅)⧵𝐽(𝑅) of [3]. In Section 2, we discover more properties shared by both zero-divisor graph and the subgraph Γ(𝑅) of Ω(𝑅). In particular, It is shown that the core of Γ(𝑅) is a union of triangles and rectangles, while a vertex in Γ(𝑅) is either an end vertex or a vertex in the core. For any nonlocal ring 𝑅, it is shown that the chromatic number of the graph Γ(𝑅) is identical with the number of maximal ideals of 𝑅. In Section 3, we introduce a new graph Γ_𝑟(𝑅) on the vertex set {𝑅𝑥∣𝑥∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅))}.(1)

This graph is in fact a retract of the graph Γ(𝑅) and is thus simpler than the graph Γ(𝑅) in general, but we will show that they share many common properties and invariants.

Jinnah and Mathew in [6] studied the problem of when a co-maximal graph Ω(𝑅) is a split graph, and they determined all rings 𝑅 with the property. In Section 4, we give an alternative proof to their main [6, Theorem 2.3]. In the co-maximal graph Ω(𝑅), each unit 𝑢 of 𝑅 is adjacent to all vertices of the graph while an element of 𝐽(𝑅) only connects to units of 𝑅. Temporally, we say 𝑢 is in the center of the graph Γ(𝑅). Related to the co-maximal relation, there is the concept of rings with stable range one. Recall that a ring 𝑅 (which needs not be commutative) has one in its stable range, if for any 𝑥,𝑦 with 𝑅𝑥+𝑅𝑦=𝑅, there is an element 𝑡 such that 𝑥+𝑡𝑦 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 𝑅 has one in its stable range if and only if for any pair of adjacent vertices 𝑥,𝑦 in the co-maximal graph Ω(𝑅), the additive coset 𝑥+𝑅𝑦 (and 𝑦+𝑅𝑥) has at least one element in the center of the graph Ω(𝑅).

Throughout this paper, all rings are assumed to be commutative with identity. For a ring 𝑅, let 𝑈(𝑅) be the set of invertible elements of 𝑅 and 𝐽(𝑅) the Jacobson radical of 𝑅. 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 𝐾 and a complete (resp., discrete) graph with 𝑛 vertices by 𝐾_𝑛 (resp., 𝐷_𝑛). A subset 𝐾 of the vertex set of 𝐺 is called a clique if any two distinct vertices of 𝐾 are adjacent; the clique number 𝜔(𝐺) of 𝐺 is the least upper bound of the size of the cliques. Similarly, we denote by 𝐾_𝑚,𝑛 the complete bipartite graph with two partitions of sizes 𝑚,𝑛, respectively. Recall that 𝐻 is called a spanning subgraph of a graph 𝐺 if 𝑉(𝐺)=𝑉(𝐻) and 𝑎∼𝑏 in 𝐻 implies 𝑎∼𝑏 in 𝐺 for all distinct vertices of 𝐺, where 𝑎∼𝑏 means that 𝑎≠𝑏 and 𝑎 is adjacent to 𝑏. Recall that a cycle in a graph is a path 𝑣₁∼𝑣₂∼⋯∼𝑣_𝑛 together with an additional edge 𝑣_𝑛∼𝑣₁ (𝑛≥3). For a simple graph 𝐺 and a nonempty subset 𝑆 of 𝑉(𝐺), there is the subgraph induced on 𝑆: the vertex set is 𝑆 and the edge set is 𝑥∼𝑦∣𝑥≠𝑦∈𝑆,andthereisanedge𝑥∼𝑦inthegraph𝐺.(2)

A discrete induced subgraph of a graph 𝐺 is also called an independent subset of 𝐺. For other notations not mentioned in the paper, one can refer to [7, 8].

2. The Subgraph Γ(𝑅)

As noted by Maimani et al. in [3], the main part of the co-maximal graph Ω(𝑅) is the subgraph Γ(𝑅) induced on the vertex subset 𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)). In fact, we have the following facts. (O1) A vertex in 𝑈(𝑅) is adjacent to every vertex of Ω(𝑅), while an element of 𝐽(𝑅) only connects to units of 𝑅. In fact, there is a sequential sum decomposition Ω(𝑅)=𝐽(𝑅)+𝑈(𝑅)+Γ(𝑅).(3) (O2) Γ(𝑅) is empty if and only if 𝑅 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 Γ(𝑅) is connected with diameter less than or equal to three.

Proof. For any 𝑎∈𝑅, set 𝑆_𝑎={𝔪∈Max(𝑅)∣𝑎∈𝔪}. Then for each 𝑎∉𝐽(𝑅), 𝑆_𝑎⊂Max(𝑅). For distinct 𝑎,𝑏∈𝑅, we claim(1)𝑎𝑏∈𝐽(𝑅) if and only if Max(𝑅)=𝑆_𝑎∪𝑆_𝑏,(2)𝑅𝑎+𝑅𝑏=𝑅 if and only if 𝑆_𝑎∩𝑆_𝑏=∅.
Now for distinct 𝑎,𝑏∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)), if 𝑎𝑏∉𝐽(𝑅), then there exists 𝑥∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)) such that 𝑅𝑎𝑏+𝑅𝑥=𝑅. Then clearly there is a path 𝑎∼𝑥∼𝑏 in Γ(𝑅) and hence 𝑑(𝑎,𝑏)≤2. If 𝑎𝑏∈𝐽(𝑅), then take any 𝑦∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)) such that 𝑅𝑎+𝑅𝑦=𝑅. We claim that 𝑏𝑦∉𝐽(𝑅) and it will follow that 𝑑(𝑎,𝑏)≤𝑑(𝑎,𝑦)+𝑑(𝑦,𝑏)≤3. In fact, assume to the contrary that 𝑏𝑦∈𝐽(𝑅). Then we have 𝑆_𝑏∪𝑆_𝑦⧵𝑆_𝑏=𝑆_𝑏∪𝑆_𝑦=Max(𝑅)=𝑆_𝑏∪𝑆_𝑎⧵𝑆_𝑏.(4)
It follows that (𝑆_𝑦⧵𝑆_𝑏)=𝑆_𝑎⧵𝑆_𝑏=∅ since 𝑆_𝑎∩𝑆_𝑦=∅. Then 𝑏∈𝐽(𝑅), a contradiction. This completes the proof.

By [4, Theorem 3.9(1)], the clique number 𝜔(Γ(𝑅)) is infinite whenever 𝐽(𝑅)=0 and the ring 𝑅 is indecomposable. It could be used to sharpen [3, Theorem 2.2], as the following theorem shows.

Theorem 2. For any non-local ring 𝑅, let 𝐺=Γ(𝑅). Then the following statements are equivalent.(1)𝐺 is a bipartite graph.(2)𝐺 is a complete bipartite graph.(3)𝑅 has exactly two maximal ideals.(4)𝑅/𝐽(𝑅)≅𝕂₁×𝕂₂, where each 𝕂_𝑖 is a field.

Proof. (3)⇒(2): Assume that 𝔪₁,𝔪₂ are the maximal ideals of 𝑅. Then Γ(𝑅) is a complete bipartite graph with two partitions 𝔪₁⧵𝔪₂ and 𝔪₂⧵𝔪₁.
(2)⇒(3): Assume that 𝐺 is a complete bipartite graph with vertex partition 𝑉(𝐺)=𝑉₁∪𝑉₂. Then for any maximal ideal 𝔪, 𝔪⧵𝐽(𝑅) is entirely contained in a single partition. Assume 𝔪_𝑖⧵𝐽(𝑅)⊆𝑉_𝑖. If 𝑅 has a third maximal ideal 𝔫, then 𝔫⧵𝐽(𝑅)⊆𝑉_𝑖 for some 𝑖. This is impossible since 𝔫+𝔪_𝑖=𝑅.
(2)⇒(1) and (4)⇒(3): Clear.
(1)⇒(3): Clearly, there is no loss to assume 𝐽(𝑅)=0.
If 𝑅 has only a finite number of maximal ideals, say 𝔪_𝑖, 1≤𝑖≤𝑟, set 𝔫_𝑖=𝔪_𝑖⧵(∪_𝑗≠𝑖𝔪_𝑗). Then 𝔫_𝑖≠∅, and each vertex in 𝔫_𝑖 is adjacent to all verices in 𝔫_𝑗. This implies 𝜔(𝐺)≥|Max(𝑅)| and hence 𝜔(𝐺)=|Max(𝑅)|, when 𝑅 has only a finite number of maximal ideals. If further 𝐺 is a bipartite graph, then 𝑟=2; that is, 𝑅 has exactly two maximal ideals.
In the following, assume that 𝑅 has infinitely many maximal ideals, and we proceed to prove 𝜔(𝐺)=∞. Assume that 𝑅 has a nontrivial idempotent 𝑒. Then 𝑅=𝑒𝑅×(1−𝑒)𝑅. If 𝑒𝑅 has no nontrivial idempotent element, then 𝜔(Γ(𝑒𝑅))=∞ and hence 𝜔(𝐺)=∞. Then assume that both 𝑒 and 1−𝑒 are nonprimitive idempotents. By induction, for any integer 𝑠≥1, there exist nontrivial corner rings 𝑅_𝑠,𝑖 of 𝑅 such that 𝑅=∏^𝑠_𝑗=1𝑅_𝑠,𝑗. Set 𝑓₁=()𝑓0,1,…,1,…,₂=()𝑓1,0,1,…,1,…,_𝑠=(1,…,1,0).(5)
Clearly {𝑓_𝑗}_{1≤𝑗≤𝑠} is a clique in Γ(𝑅), and thus 𝜔(𝐺)=∞.
(3)⇒(4): 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 𝜔(Γ(𝑅))=|Max(𝑅)| whenever 𝑅 is not a local ring.

Recall that a ring 𝑅 is called an exchange ring if the left module _𝑅𝑅 has the exchange property, see [9] and the included references for details. Recall that idempotents can be lifted modulo every ideal of an exchange ring 𝑅. 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 𝑅 with all idempotents central in 𝑅, 𝑅 is clean if and only if 𝑅 is an exchange ring. For commutative clean rings 𝑅, we have the following.

Corollary 3. For any commutative non-local exchange ring 𝑅, let 𝐺=Γ(𝑅). Then 𝐺 is a bipartite graph if and only if 𝐺 is a complete bipartite graph, if and only if 𝑅≅𝑅₁×𝑅₂, where each 𝑅_𝑖 is a local ring.

A graph 𝐺 is called totally disconnected if the edge set 𝐸(𝐺) is empty. By Theorem 1, we have the following observation and hence Theorem 4.

(O3) Γ(𝑅) is totally disconnected if and only if it is an empty graph, which is true if and only if 𝑅 is a local ring.

Theorem 4 (see [3, 4]). For a ring 𝑅, let 𝐺=Γ(𝑅). Then the following statements are equivalent.(1)A star graph is a spanning subgraph of 𝐺, that is; 𝐺 has at least two vertices, and there exists a vertex in 𝐺 which is adjacent to every other vertex.(2)𝐺 is a tree; that is, 𝐺 is nonempty, connected and contains no cycles.(3)𝐺 is a star graph.(4)𝑅 is isomorphic to ℤ₂×𝔽 for some field 𝔽.

Proof. (1)⇔(3)⇔(4): This is contained in [3, Corollary 2.4(2)].
(2)⇔(4): This is contained in [4, Theorem 3.5, Corollary 3.6].

Corollary 5. For a finite simple graph 𝐺 with |𝐺|≥2, assume 𝐺=Γ(𝑅) for some ring 𝑅. Then the following are equivalent:(1)A star graph is a spanning subgraph of 𝐺.(2)𝐺 is a tree; that is, 𝐺 contains no cycle.(3)𝐺=𝐾_{1,𝑝𝑛−1} for some prime number 𝑝 and some positive integer 𝑛.

The works of [3, 4] show that the graph Γ(𝑅) 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 𝐺 is always a union of triangles and squares, and a vertex in 𝐺 is either an end vertex or a vertex of the core. Recall that the core of a graph 𝐺 is by definition the subgraph induced on all vertices of cycles of 𝐺. In the final part of this section, we will show that the graph Γ(𝑅) has the same property.

Lemma 6. For any path 𝑎∼𝑥∼𝑏 in the graph Γ(𝑅), if 𝑎𝑏+𝑥 is not a unit of 𝑅, then the path is contained in a subgraph isomorphic to 𝐾₁+𝐾₂+𝐾₁.

Proof. The given condition implies 𝑅𝑥+𝑅𝑎𝑏=𝑅. Hence 𝑎𝑏∉𝐽(𝑅) and in particular, 𝑎𝑏+𝑥≠𝑥. Furthermore, we have 𝑅(𝑎𝑏+𝑥)+𝑅𝑎=𝑅=𝑅(𝑎𝑏+𝑥)+𝑅𝑏.(6)
Now assume 𝑎𝑏+𝑥∉𝑈(𝑅). Then 𝑎𝑏+𝑥∉𝑈(𝑅)∪𝐽(𝑅). If 𝑎𝑏+𝑥∉{𝑎,𝑏}, then there is a subgraph 𝐾₁+𝐾₂+𝐾₁ in Γ(𝑅) which contains the path 𝑎∼𝑥∼𝑏, see Figure 1. Since 𝑅(𝑎𝑏+𝑥)+𝑅𝑎=𝑅=𝑅(𝑎𝑏+𝑥)+𝑅𝑏, it follows that 𝑎𝑏+𝑥∉{𝑎,𝑏}, and this completes the proof.

Figure 1 

Lemma 6.

Lemma 7. If a vertex 𝑥 of   Γ(𝑅) is in a cycle of five vertices, then 𝑥 is in either a triangle or a rectangle.

Proof. Assume 𝑎∼𝑥∼𝑏∼𝑐∼𝑑∼𝑎 is a cycle in Γ(𝑅), and 𝑥 is not in any triangle, see Figure 2. We proceed to verify that 𝑥 is in a square.
In fact, if 𝑥𝑣≠𝑥 for some 𝑣∈𝑈(𝑅), then there is a square 𝑎∼𝑥∼𝑏∼𝑣𝑥∼𝑎 in Γ(𝑅). Thus in the following, we assume 𝑢𝑦=𝑦,forall𝑢∈𝑈(𝑅) where 𝑦∈{𝑎,𝑥,𝑏}. Then by Lemma 6, we can assume further 𝑎𝑏+𝑥=1,𝑥𝑐+𝑏=1,𝑥𝑑+𝑎=1.(∗)
Note that 𝑥𝑑 is adjacent to 𝑎, and 𝑥𝑐 is adjacent to 𝑏 in Γ(𝑅). If 𝑥𝑑=𝑥, then by (∗) we have 𝑎=𝑎𝑏. Then there is a rectangle 𝑎∼𝑥∼𝑏∼𝑑∼𝑎 in Γ(𝑅). Therefore we may assume 𝑥𝑑∉{𝑥,𝑑},𝑥𝑐∉{𝑥,𝑐}.
Since 𝑑(𝑥𝑐,𝑥𝑑)≤3, we can assume there is a path 𝑥𝑐∼𝑒∼𝑓∼𝑥𝑑. If 𝑒≠𝑏, then there is a rectangle 𝑥∼𝑒∼𝑥𝑐∼𝑏∼𝑥. If 𝑒=𝑏, then 𝑓∉{𝑥,𝑎} and hence by (∗) there is a cycle 𝑎∼𝑥∼𝑏∼𝑓∼𝑥𝑑∼𝑎. Then there is a rectangle 𝑎∼𝑥∼𝑓∼𝑥𝑑∼𝑎 and a triangle 𝑏∼𝑥∼𝑓∼𝑏. This completes the verification.

Figure 2 

Lemma 7.

Lemma 8. Let 𝐺=Γ(𝑅) and assume that 𝐺 contains a cycle. Then for any path 𝑎∼𝑥∼𝑏 in the core of 𝐺, 𝑥 is in a cycle 𝐶_𝑛 with 3≤𝑛≤5.

Proof. Assume ⋯∼𝑓∼𝑎∼𝑥∼𝑏∼𝑔∼⋯(7) is a cycle in Γ(𝑅). We proceed to verify that 𝑥 is in a cycle 𝐶_𝑛 with 3≤𝑛≤5. Since 𝑅𝑎𝑏+𝑅𝑓𝑏=𝑅𝑏, we have 𝑅𝑏=𝑅𝑎𝑏+(𝑅𝑎+𝑅𝑥)𝑓𝑏=𝑅𝑎𝑏+𝑅𝑥𝑓𝑏.(8)
Then 𝑅𝑏+𝑅(𝑥−𝑥𝑓𝑏)=𝑅𝑎𝑏+𝑅𝑥=𝑅. It follows that 𝑥−𝑥𝑓𝑏∉𝑈(𝑅)∪𝐽(𝑅). Consider 𝑥𝑓𝑏. If 𝑥𝑓𝑏=0, then it follows that 𝑅𝑏=𝑅𝑎𝑏+𝑅𝑥𝑓𝑏=𝑅𝑎𝑏. In this case, there is a rectangle 𝑎∼𝑥∼𝑏∼𝑔∼𝑎. In the following, assume 𝑥𝑓𝑏≠0, that is, 𝑥−𝑥𝑓𝑏≠𝑥, and let 𝑦=𝑥−𝑥𝑓𝑏. Without loss of generality, assume 𝑑(𝑦,𝑓)=3 and 𝑦∼𝑑∼𝑒∼𝑓 is a path from 𝑦 to 𝑓. If further 𝑑≠𝑏, then there is a rectangle 𝑏∼𝑥∼𝑑∼𝑦∼𝑏. If 𝑑=𝑏, then there is a cycle 𝐶₅∶ 𝑓∼𝑎∼𝑥∼𝑏∼𝑒∼𝑓. This completes the proof.

Theorem 9. For a ring 𝑅, let 𝐺=Γ(𝑅) and assume that 𝐺 contains a cycle. Then the following hold.(1)The core of 𝐺 is a union of triangles and rectangles, and the girth of 𝐺 is less than or equal to four.(2)Every vertex of 𝐺 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 𝑎∼𝑥∼𝑏 be a path and assume to the contrary that 𝑥 is not in the core of Γ(𝑅). Since 𝐺 contains a cycle, by Theorem 1, we can assume that 𝑎∼𝑥∼𝑏∼𝑐 is a path, where both 𝑏 and 𝑐 are in the core. Clearly, 1−𝑎𝑐∈𝑈(𝑅). By Lemma 6, both 𝑎𝑏+𝑥 and 𝑥𝑐+𝑏 are units of 𝑅. By assumption on 𝑥, we can assume further 𝑎𝑏+𝑥=1,𝑥𝑐+𝑏=1.(9)
If 𝑥𝑐=𝑥, then we have 𝑐𝑥+𝑎𝑏=𝑥+𝑏=1, a contradiction. Then 𝑥𝑐≠𝑥 and 𝑏(1−𝑎𝑐)=1−𝑐, where 1−𝑎𝑐∈𝑈(𝑅). Thus we can replace 𝑏 by 1−𝑐. In a similar way, replace 𝑥 by 1−𝑎. Now in Γ(𝑅) there is a path 𝑎∼(1−𝑎)∼(1−𝑐)∼𝑐, where both 𝑐 and 1−𝑐 are in the core of 𝐺. Certainly, there is a vertex 𝑑∉{𝑐,1−𝑐} in the core such that 1−𝑐 is adjacent to 𝑑. Then a similar argument shows 1−𝑐=𝑣(1−𝑑) for some 𝑣∈𝑈(𝑅). Since 1−𝑑∉{1−𝑐,𝑐}, and the distance from 𝑎 to 1−𝑑 is at most three, we must have a triangle (1−𝑐)∼𝑑∼(1−𝑑)∼(1−𝑐) in the graph 𝐺. Then 1−𝑐=𝑣(1−𝑑) implies (1−𝑑)∈𝑈(𝑅), a contradiction. This completes the proof.

3. A Retract Γ_𝑟(𝑅) of Γ(𝑅)

For simple graphs 𝐺 and 𝐻, recall that 𝐺→−^𝜑𝐻 if there exists a map 𝜑∶𝐺→𝐻 such that for distinct 𝑢,𝑣∈𝑉(𝐺), 𝑢∼𝑣 in 𝐺 implies 𝜑(𝑢)≠𝜑(𝑣) and 𝜑(𝑢)∼𝜑(𝑣) in 𝐻. Such a map 𝜑 is called a graph homomorphism from 𝐺 to 𝐻. If 𝐻 is a subgraph of 𝐺, and there is a graph homomorphism 𝐺→−^𝜑𝐻, where the restriction of 𝜑 on 𝐻 is an identity, then 𝐻 is called a retract of 𝐺, see Figure 3 for an example. If 𝐺 has no proper retract, then 𝐺 is called a core graph (e.g., 𝐾_𝑛 is a core graph.). Note that the core of a graph needs not to be a core graph. Recall that the chromatic number 𝜒(𝐺) of 𝐺 is the least positive integer 𝑟 such that there exists a graph homomorphism 𝐺→−^𝜑𝐾_𝑟. It is the least number of colors needed for coloring the vertices of 𝐺 in such a way that no two adjacent vertices have a same color. The girth of a graph 𝐺, denoted by 𝑔(𝐺), is the length of the minimal cycle in 𝐺.

Figure 3 

Retract for ring ℤ12.

In this section, we introduce a new graph which is a retract of Γ(𝑅) and we study this new graph.

Definition 10. For a ring 𝑅 and any 𝑥∈𝑅, let 𝑥=𝑅𝑥. Construct a simple graph in the following and denote it as Γ_𝑟(𝑅)=𝐺: 𝑉(𝐺)={(𝑈(𝑅)(𝑅,𝐸(𝐺)=𝑥∣𝑥∈𝑅⧵∪𝐽))}𝑥,𝑦∣𝑥≠𝑦∈𝑉(𝐺),and𝑅𝑥+𝑅𝑦=𝑅.(10)

Clearly, this graph has less vertices and less edges than that of the graph Γ(𝑅) in general, see Figure 3. More precisely, we have the following.

Proposition 11. (1) For a ring 𝑅, the graph Γ_𝑟(𝑅) is a retract of Γ(𝑅).
(2) For any ideal 𝐼 of 𝑅 contained in 𝐽(𝑅), Γ(𝑅/𝐼) is a retract of Γ(𝑅).
Furthermore, we have the following.
(3) If the girth of Γ(𝑅) is three, then so is the girth of Γ_𝑟(𝑅).
(4) Γ_𝑟(𝑅) is connected with diameter less than or equal to three.
(5) 𝜔(Γ_𝑟(𝑅))=𝜔(Γ(𝑅)) and 𝜒(Γ_𝑟(𝑅))=𝜒(Γ(𝑅)).
(6) Γ_𝑟(𝑅) and Γ(𝑅) are homomorphically equivalent to the same unique core graph.

Proof. (1) Clearly, 𝜑∶𝑥↦𝑥 is a surjective graph homomorphism. For any 𝑥∈𝑉(Γ_𝑟(𝑅)), fix a vertex 𝑦_𝑥 in 𝑅𝑥 to obtain a subgraph of Γ(𝑅). Then the graph Γ_𝑟(𝑅) is isomorphic to the subgraph of Γ(𝑅). Thus Γ_𝑟(𝑅) is a retract of the graph Γ(𝑅). In a similar way, one checks (2). All the remaining results follow from (1). We check (3) and (5) in the following.
(3) The result is clear by the definition of Γ_𝑟(𝑅). In fact, if 𝐺→𝐻 and 𝐺 has odd girth, then it is known that 𝑔(𝐺)=𝑔(𝐻), since any cycle with odd length is a core graph.
(5) Since a composition of graph homomorphisms is still a graph homomorphism, clearly 𝜒(Γ_𝑟(𝑅))≥𝜒(Γ(𝑅)). On the other hand, since Γ_𝑟(𝑅) is isomorphic to a subgraph of Γ(𝑅), for any graph homomorphism 𝜓∶Γ(𝑅)→𝐾_𝑠, the restriction 𝜓|∶Γ_𝑟(𝑅)→𝐾_𝑠 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 [12] for details.

By Theorem 2 and Proposition 11, we have the following.

Corollary 12. (1) For any non-local ring 𝑅, 𝜒(Γ_𝑟(𝑅))≥|Max(𝑅)|.
(2) Γ_𝑟(𝑅) is a complete bipartite graph if and only if 𝑅 has exactly two maximal ideals, which is true if and only if Γ_𝑟(𝑅) is a bipartite graph.

By [4, Corollary 3.8(1)], if 𝐻=Γ(𝑅) contains a cycle, then the girth 𝑔(𝐻)≤4. Compared with Proposition 11(3), we have the following example for the 𝑔(𝐻)=4 case (see Figure 3).

Example 13. Consider the ring 𝑅=ℤ₁₂. For this ring, 𝑔(Γ(𝑅))=4, while 𝑔(Γ_𝑟(𝑅))=∞. We draw the two graphs in Figure 3.
Recall a convenient construction from graph theory, the sequential sum𝐺=𝐺₁+𝐺₂+⋯+𝐺_𝑟(11) of a sequence of graphs 𝐺₁,𝐺₂,…,𝐺_𝑟 with 𝑉(𝐺_𝑖)∩𝑉(𝐺_𝑗)=∅ for all 𝑖≠𝑗. The vertex set of 𝐺 is ∪^𝑟_𝑖=1𝑉(𝐺_𝑖) and the edge set of 𝐺 is 𝐸∪(∪^𝑟_𝑖=1𝐸(𝐺_𝑖)), where 𝐸=𝑣_𝑖∼𝑣_𝑖+1∣𝑣_𝑖∈𝑉𝐺_𝑖,𝑣_𝑖+1∈𝑉𝐺_𝑖+1,1≤𝑖≤𝑟−1.(12)
We illustrate the construction in Figure 4 for the sequence of graphs 𝐾₁,𝐾₁,𝐻, where 𝐻 is a triangle 𝐾₃ together with three end vertices adjacent to distinct vertices.

Figure 4 

Sequential sum.

Theorem 14. Let 𝑅 be any commutative ring that is not a local ring, and let 𝐺=Γ(𝑅). Then the following numbers are identical:(1)the chromatic number 𝜒(𝐺),(2)the clique number 𝜔(𝐺),(3)the cardinal number of Max(𝑅),(4)𝜒(Γ_𝑟(𝑅)), (5)𝜔(Γ_𝑟(𝑅)).

Proof. First, recall from [2, Theorem 2.3] that 𝜒(Ω(𝑅))=|Max(𝑅)|+|𝑈(𝑅)|.(13)
Then consider the following decomposition of the co-maximal graph into sequential sums of three subgraphs: Ω(𝑅)=𝐽(𝑅)+𝑈(𝑅)+Γ(𝑅),(14) where 𝐽(𝑅) is a discrete subgraph while 𝑈(𝑅) is a complete subgraph. Since |𝐽(𝑅)|≤|𝑈(𝑅)|, we have 𝜒(Ω(𝑅))=𝜒(Γ(𝑅))+|𝑈(𝑅)|.(15)
By [4, Theorem 3.9(2)], 𝜔(Γ(𝑅))=|Max(𝑅)|. Then 𝜒(Γ(𝑅))=|Max(𝑅)|=𝜔(Γ(𝑅)).(∗∗)
By Proposition 11, we have 𝜒(𝐺)=𝜒(Γ_𝑟(𝑅))≥𝜔(Γ_𝑟(𝑅))=𝜔(𝐺), and the result follows from (∗∗).

Now we study the interplay between the graph structure of Γ_𝑟(𝑅) and the algebraic property of 𝑅. First we have the following

Proposition 15. For a ring 𝑅, let 𝐺=Γ_𝑟(𝑅). Then the following are equivalent:(1)A star graph is a spanning subgraph of 𝐺.(2)𝐺 is a star graph.(3)𝑅≅𝔽×𝑇, where 𝔽 is a field and 𝑇 is a local ring.

Proof. We only need to prove (1)⇒(3). Assume 𝑥 is adjacent to every other vertex in Γ_𝑟(𝑅). Then 𝑥=𝑥². Assume 𝑥=𝑟𝑥². Then 𝑟𝑥=(𝑟𝑥)² and 𝑥=𝑟𝑥. So we can assume at the start that 𝑥 is a nontrivial idempotent. Let 𝑅𝑥=𝑇,𝔽=𝑅(1−𝑥). Then 𝑇 is a maximal ideal of 𝑅 and 𝑅=𝔽×𝑇. Thus 𝔽 is a field. Now 𝐽(𝑅)=𝐽(𝑇), and for any 𝑦∈𝑇⧵𝐽(𝑇), we have 𝑇=𝑇𝑦. Thus 𝑇 is a local ring with the maximal ideal 𝐽(𝑅).

Note that for a finite local ring (𝑇,𝔪), if 𝔪=𝑇𝑥₁∪⋯∪𝑇𝑥_𝑟 in which 𝑇𝑥_𝑖≠𝑇𝑥_𝑗, then Γ_𝑟(𝑅)=𝐾_1,𝑟.

Corollary 16. For any commutative ring 𝑅, diam(Γ_𝑟(𝑅))=1 if and only if 𝑅≅𝔽₁×𝔽₂, where each 𝔽_𝑖 is a field.

Proof. If 𝑅≅𝔽₁×𝔽₂, then clearly diam(Γ_𝑟(𝑅))=1. Conversely, assume diam(Γ_𝑟(𝑅))=1. Then Γ_𝑟(𝑅) is a complete graph. By Proposition 15, we have 𝑅≅𝔽×𝑇, where 𝔽 is a field and 𝑇 is a local ring. If 𝑇 is not a field, then take any nonzero 𝑥∈𝐽(𝑇). Then (1,𝑥)∈𝑅⧵𝑈(𝑅)⧵𝐽(𝑅), and (1,0) is not adjacent to (1,𝑥) in Γ_𝑟(𝑅), a contradiction. This completes the proof.

The following result follows from Corollary 12, Proposition 15, and the proof of [3, Lemma 3.2, Proposition 3.3],

Proposition 17. For any commutative non-local ring 𝑅, diam(Γ_𝑟(𝑅))=2 if and only if one of the following conditions holds. (1)𝐽(𝑅) is a prime ideal of 𝑅.(2)𝑅 has exactly two maximal ideals, and 𝑅≇𝔽₁×𝔽₂ for any fields 𝔽_𝑖.

Corollary 18. For any commutative non-local ring 𝑅, the graphs Γ(𝑅) and Γ_𝑟(𝑅) have the same diameter if and only if 𝑅≇𝔽₁×𝔽₂ for any fields 𝔽_𝑖 but 𝔽₁=𝔽₂=ℤ₂.

By [4, Theorem 3.9(2)], Theorem 14, and Corollary 16, if diam(Γ_𝑟(𝑅))=2 (resp., diam(Γ(𝑅))=2), then either Γ_𝑟(𝑅) (resp., Γ(𝑅)) is a complete bipartite graph or its clique number is infinite.

At the end of this section, we pose the following problem.

Question 1. Which rings 𝑅 have the property that Γ(𝑅) is a generalized split graph? Which rings 𝑅 have the property that Γ_𝑟(𝑅) is a (generalized) split graph?

Recall from [13] that a simple graph 𝐺 is a generalized split graph if 𝑉(𝐺)=𝐾∪𝐷,𝐾∩𝐷=∅,(16) where the induced subgraph on 𝐾 (resp., on 𝐷) is a core graph (resp., a discrete graph). Note that Γ(ℤ₁₂) is not a generalized split graph, while Γ_𝑟(ℤ₁₂) is a split graph.

4. A Ring 𝑅 Whose Comaximal Graph Ω(𝑅) Is a Split Graph

Throughout this section, assume that 𝐺 is a split graph, that is, 𝐺 is simple and connected with 𝑉(𝐺)=𝐾∪𝐷, where 𝐾∩𝐷=∅ and the induced subgraph on 𝐾 (resp., on 𝐷) is a complete (resp., discrete) graph, see [13] and the included references there. For the split graph 𝐺, we always assume that 𝐷 is a maximal such independent subset. Under this assumption, a complete graph 𝐾_𝑛 is a split graph with |𝐾|=𝑛−1,|𝐷|=1.

Lemma 19. For a commutative ring 𝑅, let 𝐺 be the co-maximal graph Ω(𝑅). If 𝐺 is a split graph with 𝑉(𝐺)=𝐾∪𝐷, then we have the following.(1)For any proper ideal 𝐼 of 𝑅, |𝐼∩𝐾|≤1.(2)𝔪∩𝔫∩𝐾={0} holds for distinct maximal ideals 𝔪,𝔫 of 𝑅.(3)If 𝑅 is isomorphic to neither ℤ₂ nor ℤ₂×ℤ₂, then |𝐾|≥|Max(𝑅)|+1. Also, |𝐾|=|Max(𝑅)| if and only if 𝑅=ℤ₂ or 𝑅≅ℤ₂×ℤ₂.

Proof. (1) Clear.
(2) Assume to the contrary that there exists some nonzero 𝑢∈𝔪∩𝔫∩𝐾. Since 𝔪+𝔫=𝑅, we have 𝑥∈𝔪,𝑦∈𝔫 such that 𝑥+𝑦=1. Clearly, 𝑥≠𝑦 and hence we can assume 𝑥∈𝐾. Then it follows from Lemma 19(1) that 𝑥=𝑢∈𝔫, a contradiction.
(3) If 𝑅 is a local ring, then either 𝑅≅ℤ₂ or 𝑅 has at least two units by assumption. The result holds in this case. In the following, we assume that 𝑅 is non-local. If there is a maximal ideal 𝔪 with 𝔪⊆𝐷, then by the proof of [6, Theorem 2.1], 𝑅≅ℤ₂×𝔽 for some field 𝔽. In this case, |𝐾|≥|Max(𝑅)| and, |𝐾|=|Max(𝑅)| if and only if 𝔽=ℤ₂. If no maximal ideal is contained in 𝐾, then each maximal ideal has exactly one vertex in 𝐾 by (1). By (2), we have |𝐾|≥|Max(𝑅)|+1.

We remark that Lemma 19(3) is the best possible. For instance, |𝐾|=|Max(𝑅)|+1 holds for the local ring ℤ₄ and 𝑍₂×𝑍₂×𝑍₂. For 𝑅=𝑍₂×𝑍₂×𝑍₂, we draw its co-maximal graph Ω(𝑅) in Figure 4, in which Ω(𝑅)=𝐾₁+𝐾₁+𝐻.

The following is a result of [6].

Lemma 20 (see [6, Theorem 2.1]). For a commutative ring 𝑅 which is non-local, if the co-maximal graph Ω(𝑅) is a split graph and 𝑅≇ℤ₂×𝔽 for any field 𝔽, then the characteristic of 𝑅 is two, 𝑅 has exactly three maximal ideal 𝔪_𝑖, and 𝐾∩𝔪_𝑖={𝑥_𝑖} for all 𝑖, where each 𝑥_𝑖 is idempotent and for every invertible element 𝑢 of 𝑅, 𝑢𝑥_𝑖=𝑥_𝑖.

We now give an alternative proof to the main result of [6].

Theorem 21 (see [6, Theorem 2.3]). For a commutative ring 𝑅, the co-maximal graph Ω(𝑅) is a split graph if and only if 𝑅 is one of the following: a local ring, ℤ₂×ℤ₂×ℤ₂, ℤ₂×𝔽 for some field 𝔽.

Proof. We only need to prove the necessary part. Assume that 𝑅 is not a local ring and 𝑅≇ℤ₂×𝔽 for any field 𝔽. Let 𝐺=Ω(𝑅). Assume further that the co-maximal graph 𝐺 is a split graph with 𝑉(𝐺)=𝐾∪𝐷. By Lemma 20, 𝑅 has exactly three maximal ideals 𝔪_𝑖 (1≤𝑖≤3) such that 𝔪_𝑖∩𝐾={𝑥_𝑖}, where 𝑥²_𝑖=𝑥_𝑖 and 𝑢𝑥_𝑖=𝑥_𝑖 hold for every 𝑢∈𝑈(𝑅). Also, 𝑅 has characteristic 2. For any 1≤𝑖≠𝑗≤3, assume 1=𝑟𝑥_𝑖+𝑠𝑥_𝑗 with 𝑟𝑥_𝑖∈𝐾. By Lemma 19, we have 𝑥_𝑖=𝑟𝑥_𝑖 and hence 1−𝑥_𝑖=𝑠𝑥_𝑗. It follows that (1−𝑥_𝑖)(1−𝑥_𝑗)=0. Note that 1+𝑥₁𝑥₂𝑥₃∈𝑈(𝑅) since 𝑥₁𝑥₂𝑥₃∈𝐽(𝑅), and it follows that 𝑥₁𝑥₂𝑥₃=0. Then (1−𝑥_𝑖)(1−𝑥_𝑗)=0 implies 𝑥_𝑘+𝑥_𝑖𝑥_𝑘+𝑥_𝑗𝑥_𝑘=0 for any rearrangement 𝑖,𝑗,𝑘 of 1,2,3. Therefore 𝑥₁+𝑥₂+𝑥₃=2⋅𝑓(𝑥₁,𝑥₂,𝑥₃)=0. This shows that 1−𝑥₁,1−𝑥₂,1−𝑥₃ is a complete set of orthogonal idempotent elements of 𝑅.
Let 𝑅_𝑖=𝑅(1−𝑥_𝑖). Then 𝑅=∑³_𝑖=1𝑅_𝑖≅𝑅₁×𝑅₂×𝑅₃, where each 𝑅_𝑖 is a local ring with a unique maximal ideal 𝔫_𝑖, since 𝑅 has exactly three maximal ideals. For any 𝑟(1−𝑥₁)∈𝔫₁, we have 1+𝑟1−𝑥₁=1−𝑥₁+𝑟1−𝑥₁+1−𝑥₂+1−𝑥₃∈𝑈(𝑅).(17)
Since 𝑢𝑥₂=𝑥₂ for all 𝑢∈𝑈(𝑅), it follows that 𝑟(1−𝑥₁)⋅𝑥₂=0. Now apply the fact (1−𝑥₁)(1−𝑥₂)=0; one derives 𝑟(1−𝑥₁)=0. Hence 𝔫₁=0 and each 𝑅_𝑖 is a field. Thus 𝑅 is a direct product of three fields.
Since 𝑥₁=𝑥₂+𝑥₃, it follows that 𝔪₁=𝑅₂+𝑅₃,𝔪₂=𝑅₁+𝑅₃,𝔪₃=𝑅₁+𝑅₂.(18)
Then in the decomposition 𝑅=𝐾∪𝐷, we deduce from Lemma 19(1) that 𝐾=𝑈𝑅₁+𝑈𝑅₂+𝑈𝑅₃∪𝑥₁,𝑥₂,𝑥₃,𝐷=𝑅⧵𝐾.(19)
Now we claim that each 𝑅_𝑖 is isomorphic to ℤ₂ and hence 𝑅≅ℤ₂×ℤ₂×ℤ₂. In fact, assume 𝑅=𝑅₁×𝑅₂×𝑅₃. Then 𝐾=𝑈𝑅₁×𝑈𝑅₂×𝑈𝑅₃∪{(0,1,1),(1,0,1),(1,1,0)}.(20)
If |𝑅₁|>2, then there exists a nonzero element 1≠𝑣₁∈𝑅₁. Then both 𝑧=(𝑣₁,0,1) and 𝑒=(0,1,0) are in the independent subset 𝐷, contradicting 𝑅𝑧+𝑅𝑒=𝑅. 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 𝐺 is one of the following: (1) the sequential sum 𝐾₁+𝐾_𝑝𝑛−1+𝐾_{1,𝑝𝑛−1} for some prime number 𝑝 and (2) the sequential sum 𝐾₁+𝐾₁+𝐻 in Figure 4, where 𝐻 is a triangle 𝐾₃ 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 (𝑅,𝔪) which is not a field, assume |𝔪|=𝑝^𝑚. Then the co-maximal graph of 𝑅 is 𝐾_{𝑝𝑛−𝑝𝑚}+𝐷_𝑝𝑚, where 𝑝^𝑛≥2𝑝^𝑚 and 𝐷_𝑝𝑚 is a discrete graph with 𝑝^𝑚 vertices.

By Theorem 21 and the results of [3, 4], it is natural to ask the following question: for what rings 𝑅 is Γ(𝑅) a split graph, where the vertex set of Γ(𝑅) is 𝑅⧵(𝑈(𝑅)∪𝐽(𝑅))? Note that for distinct maximal ideals 𝔪,𝔫 (if exist) and 𝑥∈𝔪,𝑦∈𝔫, 𝑅𝑥+𝑅𝑦=𝑅 implies 𝑥,𝑦∈𝑅⧵(𝑈(𝑅)∪𝐽(𝑅)). Then a careful check to the proofs of Lemma 19, Lemma 20, and Theorem 21 shows the following.

Theorem 23. For any ring 𝑅, the following statements are equivalent.(1)The co-maximal graph Ω(𝑅) of 𝑅 is a split graph.(2)The subgraph Γ(𝑅) is either empty or a split graph.(3)𝑅 is one of the following: a local ring, ℤ₂×ℤ₂×ℤ₂, ℤ₂×𝔽 for some field 𝔽.

Note that a split graph 𝐺 is isomorphic to Γ(𝑅) for some finite ring 𝑅 if and only if either 𝐺 is a star graph 𝐾_{1,𝑝𝑛−1} for some prime number 𝑝 or 𝐺 is the triangle together with three end vertices adjacent to distinct vertices (see the graph 𝐻 in Figure 4).

Acknowledgment

This research is supported by the National Natural Science Foundation of China (Grant no. 11271250).