The Comaximal Graphs of Noncommutative Rings.

For a ring R (not necessarily commutative) with identity, the comaximal graph of R , denoted by Ω (R) , is a graph whose vertices are all the nonunit elements of R , and two distinct vertices a and b are adjacent if and only if R a + R b = R. In this paper we consider a subgraph Ω 1 (R) of Ω (R) induced by R \ U ℓ (R) , where U ℓ (R) is the set of all left-invertible elements of R. We characterize those rings R for which Ω 1 (R) \ J (R) is a complete graph or a star graph, where J (R) is the Jacobson radical of R. We investigate the clique number and the chromatic number of the graph Ω 1 (R) \ J (R) , and we prove that if every left ideal of R is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of Ω 1 (R) \ J (R). We also investigate the properties of R when Ω 1 (R) is a split graph. [ABSTRACT FROM AUTHOR]