# Higher derivations and commutativity in lattice-ordered rings

Higher derivations and commutativity in lattice-ordered rings In 1978 I. N. Herstein proved that a prime ring $$R$$ R of characteristic not two with nonzero derivation $$d$$ d satisfying $$d(x)d(y)=d(y)d(x)$$ d ( x ) d ( y ) = d ( y ) d ( x ) for all $$x,y\in R$$ x , y ∈ R is commutative, and in 1995 Bell and Daif showed that $$d(xy)=d(yx)$$ d ( x y ) = d ( y x ) implies commutativity. We extend the Bell–Daif theorem to lattice-ordered prime rings with a positive derivation satisfying the property on a one-sided $$L$$ L -ideal and interpret these conditions for higher derivations in prime $$d$$ d -rings and in semiprime $$f$$ f -rings. Our key tool is that every positive derivation nilpotent on a one-sided $$L$$ L -ideal of a semiprime $$\ell$$ ℓ -ring is zero on that ideal. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Higher derivations and commutativity in lattice-ordered rings

Positivity, Volume 18 (3) – Dec 11, 2013
15 pages

/lp/springer_journal/higher-derivations-and-commutativity-in-lattice-ordered-rings-NK2TDuHblb
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0266-0
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

PositivitySpringer Journals

Published: Dec 11, 2013

### References

• Commutativity of rings with derivations
Andima, S; Pajoohesh, H

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