, Volume 18, Issue 3, pp 603–617 | Cite as

Higher derivations and commutativity in lattice-ordered rings

  • S. Andima
  • H. PajooheshEmail author


In 1978 I. N. Herstein proved that a prime ring \(R\) of characteristic not two with nonzero derivation \(d\) satisfying \(d(x)d(y)=d(y)d(x)\) for all \(x,y\in R\) is commutative, and in 1995 Bell and Daif showed that \(d(xy)=d(yx)\) implies commutativity. We extend the Bell–Daif theorem to lattice-ordered prime rings with a positive derivation satisfying the property on a one-sided \(L\)-ideal and interpret these conditions for higher derivations in prime \(d\)-rings and in semiprime \(f\)-rings. Our key tool is that every positive derivation nilpotent on a one-sided \(L\)-ideal of a semiprime \(\ell \)-ring is zero on that ideal.


Derivation Higher derivation Lattice-ordered ring \(f\)-ring \(d\)-ring Prime ring Semiprime ring 

Mathematics Subject Classification (2000)

16N60 13N15 16U80 06F25 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsLong Island University-C.W. Post CampusBrookvilleUSA
  2. 2.Department of MathematicsMedgar Evers College of CUNYBrooklynUSA

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