Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute

Abstract

An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. Then R is said to be strongly g(x)-nil clean if every element in R is a sum of a nilpotent and a root of g(x) that commute. In this paper, we give some relations between strongly nil clean rings and strongly g(x)-nil clean rings. Various basic properties of strongly g(x) -nil cleans are proved and many examples are given.

Keywords

• nil clean ring
• strogly nil clean ring
• g(x)-nil clean ring
• strongly g(x)-nil clean ring
• 16n40
• 16u99