Nilpotency of Derivations

Chung, Lung Ock; Luh, Jiang

doi:10.4153/cmb-1983-057-5

Cited by 22 publications

(14 citation statements)

References 4 publications

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“…When R is a prime ring, m δ (R) is then the least integer m such that R δ m c = 0 for some nonzero c ∈ R. The annihilating nilpotency of δ is first investigated in [9]; we give it this name for brevity. Nilpotent derivations enjoy many interesting properties and have been studied extensively [4][5][6]9,15]. When R has a prime characteristic p 2, most results impose restrictions on p. Our aim here is to remove these restrictions.…”

Section: Introductionmentioning

confidence: 98%

Nilpotent derivations

Chuang

Lee

2005

Journal of Algebra

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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]

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Section: Introductionmentioning

confidence: 98%

Nilpotent derivations

Chuang

Lee

2005

Journal of Algebra

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“…Montgomery [5] shows that if a finite group acts on a domain then the trace map is non-zero if and only if it is non-zero on every non-zero one-sided ideal. Chung and Luh [2] show that if d is a derivation of a prime ring then d" is zero on a non-zero ideal if and only if d" is zero on the entire ring. Lanski [3] extends this result to show that if a derivation is algebraic on a non-zero ideal of a prime ring then the same polynomial in the derivation vanishes on the entire ring.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…We will embed our automorphic-differential endomorphisms in a set of polynomials which resembles a skew group ring and a Lie algebra smash product [1]. We avoid the calculations required for a fixed endomorphism [2,3] by instead exploiting the properties of endomorphisms with minimal degrees as polynomials.…”

Section: Introduction and Notationmentioning

confidence: 99%

Automorphic-differential identities in rings

Bergen¹

1989

Proc. Amer. Math. Soc.

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Abstract.Let R be a ring and f an endomorphism obtained from sums and compositions of left multiplications, right multiplications, automorphisms, and derivations. We prove several results relating the behavior of / on certain subsets of R to its behavior on all of R . For example, we prove ( 1 ) if R is prime with ideal / / 0 such that /(/) = 0, then f(R) = 0, (2) if R is a domain with right ideal X ¿ 0 such that f(X) = 0, then f(R) = 0, and (3) if R is prime and /(A") = 0 , for X a right ideal and n > 1 , then f(X) = 0 . We also prove some generalizations of these results for semiprime rings and rings with no non-zero nilpotent elements.

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“…Recently, the present authors in [5] (1), replaze x by d-2+X(a) and note that d j (a) P for all j < s. We obtain 2nd(a)d(y) fi P for all y R. (2) Replacing y by xy in (2) yields 2nd(a)xd(y) + 2nd(a)d(z)y P for all z, y R.…”

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confidence: 82%