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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 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.

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