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. 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 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 [3] 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. [3] and Wang [4], 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 [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 to denote the graph of [3]. 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 [6] 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,