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

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

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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 illumivein 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 laufenn 275/55r20 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.