ON THE GIRTH OF THE UNIT GRAPH OF A RING

Abstract

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth gr (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with gr (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr (G(R)) = 3 or 4.