DERIVATIONS AND SKEW POLYNOMIAL RINGS

Abstract. Let R be a prime ring and d a derivation of R. In the ring of additive endomorphisms of the abelian group (R,+), let S be the subring generated by aLdm, where a ∈ R and m ≥ 0 and where aL:x ∈ R 7 → ax ∈ R for a ∈ R. We compute the prime radical and minimal prime ideals of S via the skew polynomial ring R[x; d] by the surjective ring homomorphism ϕ: n∑ i=0 aix n−i ∈ R[x; d] 7→ n∑ i=0 (ai)Ldn−i ∈ S. We compute explicitly the kernel A of ϕ, the prime radical P over A and minimal prime ideals over A (Theorem 2). We obtain a necessary and sufficient condition for S to be simple, prime or semiprime (Corollary 3). As an application, let d be nilpotent. We show that the d-extension of R defined in [6] is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14).