Nonassociative rings with skew derivations and rings with associators in the nuclei

We give preliminaries. In [25] we obtain that if R is a prime ring with a s-derivation d such that sd=ds, and d(R) ⊆ N∩L or N∩M or M∩L then R is associative or d^2=2d=0. Applying the result, we have the theorem by Posner for the prime nonassociative rings. The result [29] is as follows: if R is a prime ring with a skew derivation d such that ax + d(x) ∈ G for all x in R where a ∈ Z the ring of rational integers or a ∈G∩C then R is associative or d(G) = 0 Finally we generalize Kleinfeld’s result [10].