Derivations and skew polynomial rings
Journal
Communications in Algebra
Journal Volume
35
Journal Issue
2
Pages
527-539
Date Issued
2007
Author(s)
Chuang, C.-L.
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 of ϕ, the prime radical 𝒫 over and minimal prime ideals over (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 Grzeszczuk , P. ( 1992 ). On nilpotent derivations of semiprime rings . J. Algebra 149 : 313 – 321 .[Crossref], [Web of Science ®] , [Google Scholar]) is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14).
Type
journal article