On NI Skew Polynomial Rings

Abstract

Let R be a ring with an endomorphism α and an α-derivation δ. In this article, we first compute the Jacobson radical of NI ℤ-graded rings and show that J(S) = Niℓ(S) if and only if (Formula presented.) is a ℤ-graded NI ring and J(S) ∩ S0 is nil. As a corollary we show that, J(R[x; α]) = Niℓ(R[x; α]) if and only if R[x; α] is NI and J(R[x; α]) ∩ R ⊆ Niℓ(R). If R[x, x−1; α] is NI we prove that, J(R[x, x−1; α]) = Niℓ(R[x, x−1; α]) = Niℓ*(R[x, x−1; α]) = Niℓ(R)[x, x−1; α]. We also provide necessary and sufficient conditions for a skew polynomial ring R[x; α, δ] and skew Laurent polynomial ring R[x, x−1; α] to be NI. © 2015, Copyright Taylor & Francis Group, LLC.