Nilpotent elements in skew polynomial rings

Letr be a ring with an endomorphism α and an α-derivation δ antoine studied the structure of the set of nilpotent elements in armendariz rings and introduced nil-armendariz rings. in this paper we introduce and investigate the notion of nil-(α,δ)-compatible rings. the class of nil-(α,δ)-compatible rings are extended through various ring extensions and many classes of nil-(α,δ)-compatible rings are constructed. we also prove that,if r is nil-α-compatible and nil-armendariz ring of power series type with nil(r) nilpotent,then nil(r[[x;α]]) = nil(r)[[x;α]]. we show that,if r is a nil-armendariz ring of power series type,with nil (r) nilpotent and nil-(α,δ)-compatible ring,then nil (r[x;α,δ]) = nil (r) [x; α,δ]. as a consequence,several known results are unified and extended to the more general setting. also examples are provided to illustrate our results.