NIL-CLEAN GRAPH OF SOME POLYNOMIAL RINGS

Abstract

A nil-clean element of a commutative ring R is an element which can be written as a sum of an idempotent element and a nil element of R. The graph constructed by taking all nonzero elements of a ring as the vertices, where any two vertices are adjacent if their sum is a nil-clean element of R is known as nil-clean graph. With a revised adjacency rule, as any two vertices are adjacent if their product is a nil-clean element, on the same vertex set we revised the definition of nil-clean graph. In this work we study the Nil-Clean Graph of some polynomial quotient rings and their graph theoretical properties.