On the Comaximal Graph of a Commutative Ring

Abstract Let R be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on R , Γ( R ), with vertices as elements of R , where two distinct vertices a and b are adjacent if and only if Ra + Rb = R . In this paper, we consider a subgraph Γ 2 ( R ) of Γ( R ) that consists of non-unit elements. We investigate the behavior of Γ2( R ) and Γ 2 (R)\ J (R), where J( R ) is the Jacobson radical of R . We associate the ring properties of R , the graph properties of Γ(R), and the topological properties of Max( R ). Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

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