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 a L d m , where a ∈ R and m ≥ 0 and where a L :x ∈ R ↦ 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 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 Grzeszczuk (1992) is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14).