A Note on Comaximal Graph of Non-commutative Rings

Abstract

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Let Γ₂(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:

• (a) If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n = 2.

• (b) If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R ≅ ℤ₂×F, where F is a field.

In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph.