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 is a ℤ-graded NI ring and J(S) ∩ S ₀ 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.

Key Words:

• Jacobson radical
• NI rings
• Skew Laurent polynomial rings
• Skew polynomial rings

2010 Mathematics Subject Classification:

• 16S36
• 16N20
• 16N40

ACKNOWLEDGMENT

Special thanks are due to the referee who read this article very carefully, made many useful suggestions, and recommend ℤ-graded version of Theorem 2.4, which improved the article. The author would like to thank the Banach Algebra Center of Excellence for Mathematics, University of Isfahan.

Notes

Communicated by A. Smoktunowicz.