On 2-primal differential polynomial rings

Let R be a ring with a derivation δ. In this note we show that if R is a nil-δ-compatible ring, then R is 2-primal if and only if the differential polynomial ring R[x; δ] is 2-primal if and only if Nil(R) = Nil∗(R; δ) if and only if Nil∗(R[x; δ]) = Ni(R)[x; δ] if and only if every minimal δ-prime ideal of R is completely prime. The class of nil δ-compatible rings contains properly reduced rings and δ-compatible rings, and contrary to the notion of δ-compatible 2-primal rings, nil-δ-compatible 2-primal rings extend to polynomial rings, triangular matrix rings and various ring extensions