Aequationes mathematicae

, Volume 82, Issue 1–2, pp 123–134 | Cite as

Nilpotent and invertible values in semiprime rings with generalized derivations



Let R be a semiprime ring and F be a generalized derivation of R and n ≥ 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) − yx) n is either zero or invertible for all \({x,y\in R}\), then there exists a division ring D such that either R = D or R = M 2(D), the 2 × 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) − yx) n  = 0 for all \({x,y \in I}\), then [I, I]I = 0, F(x) = ax + xb for \({a,b\in R}\) and there exist \({\alpha, \beta \in C}\), the extended centroid of R, such that (aα)I = 0 and (bβ)I = 0, moreover ((a + b)xx)I = 0 for all \({x\in I}\).

Mathematics Subject Classification (2000)

16N60 16W25 


Prime and semiprime rings generalized derivations 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.DI.S.I.A., Faculty of EngineeringUniversity of MessinaMessinaItaly

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