Nilpotent derivations

Abstract

Motivated by Grzeszczuk's paper [P. Grzeszczuk, On nilpotent derivations of semiprime rings, J. Algebra 149 (1992) 313-321], we give a detailed analysis of nilpotent derivations of semiprime rings. With this, many known results can be either generalized or deduced.  2005 Elsevier Inc. All rights reserved. IntroductionThroughout, R is always a semiprime ring and Q its symmetric Martindale quotient ring. The center C of Q is called the extended centroid of R. By a derivation of R we mean a map δ : R → R satisfying (x + y) δ = x δ + y δ and (xy) δ = x δ y + xy δ for all x ∈ R.A derivation of this form is called inner. More generally, a derivation δ of R is called X-inner if it is the restriction of an inner derivation of Q, that is, if there exists b ∈ Q such that x δ = [x, b]

Nilpotency ofq-Skew Derivations

“…The description of nilpotent generalized derivations and some related results were obtained in a recent article by Brešar et al (2004). Other related results can be found in Beidar and Brešar (2000), Lee (2000), Lee and Shiue (2000), and Chuang and Lee (2005b). In this article, we extend the result to the case of q-skew derivations.…”

“…In fact, it follows the viewpoint of Chuang and Lee (2005b) that we will handle the general q-skew case very well. Therefore, we do not record this special case.…”

Higher derivations of Ore extensions by q-skew derivations

“…This implies ν ≥ n since n i q = 0 for 0 < i < n. But n = kν ≥ ν. So ν = n. If char R = p ≥ 2 write n = νp k l, where p l. By Lemma 5, νp k l νp k q = p k l p k ≡ p l = 0 by Lucas' Lemma [4, Lemma 2] or [5,Lemma 2.6]. This implies νp k ≥ n since n i q = 0 for 0 < i < n. But n = νp k l ≥ νp k .…”

The kernel and range inclusions of integral derivations in semiprime rings

“…(See[5, Theorem 1.5].) If a ∈ Q is a C -integral element of degree m then there exist uniquely orthogonal g i ∈ B, 1 i m, with m i=1 g i = 1 such that each ag i , if g i = 0, has a fully minimal polynomial of degree i in the ring g i Q .…”