Some properties of comaximal right ideal graph of a ring

For a ring R (not necessarily commutative) with identity, the comaximal right ideal graph of R, denoted by G(R), is a graph whose vertices are the nonzero proper right ideals of R, and two distinct vertices I and J are adjacent if and only if I+J=R. In this paper we consider a subgraph G*(R) of G(R) induced by V(G(R))∖J(R), where J(R) is the set of all proper right ideals contained in the Jacobson radical of R. We prove that if R contains a nontrivial central idempotent, then G*(R) is a star graph if and only if R is isomorphic to the direct product of two local rings, and one of these two rings has unique maximal right ideal {0}. In addition, we also show that R has at least two maximal right ideals if and only if G*(R) is connected and its diameter is at most 3, then completely characterize the diameter of this graph.