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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.20 No.4

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THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2013, v.20 no.4, pp.243-249
https://doi.org/10.7468/jksmeb.2013.20.4.243
Toumi, Mohamed Ali
Toumi,, M. A. (2013). THE RANGE INCLUSION RESULTS FOR ALGEBRAIC NIL DERIVATIONS ON COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 20(4), 243-249, https://doi.org/10.7468/jksmeb.2013.20.4.243

Abstract

Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any <TEX>$x{\in}A$</TEX> there is a positive integer n = n(x) such that <TEX>$D^{n(x)}(P(x))=0$</TEX>, for all <TEX>$P{\in}\mathbb{C}[X]$</TEX> (by convention <TEX>$D^{n(x)}({\alpha})=0$</TEX>, for all <TEX>${\alpha}{\in}\mathbb{C}$</TEX>). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra A maps into N(A), where N(A) denotes the set of all nilpotent elements of A. As an application, we deduce that any nilpotent derivation on a commutative complex algebra A maps into N(A), Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.

keywords
algebraic nil derivation, d-algebra, nilpotent derivation

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