The Comaximal Graphs of Noncommutative Rings

Abstract

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