On nilpotent elements of ore extensions

Let R be an associative ring with unity, α be an endomorphism of R and δ an α-derivation of R. We introduce the notion of α-nilpotent p.p.-rings, and prove that the α-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when R is a nil-α-compatible and 2-primal ring with Nil(R) nilpotent, then Nil(R[x; α,δ]) = Nil(R)[x; α,δ]; and when R is a nil Armendriz ring of skew power series type with Nil(R) nilpotent, then Nil(R[[x; α]]) = Nil(R)[[x; α]], where Nil(R) is the set of nilpotent elements of R. These results extend existing results to a more general setting.