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.
Positivity – Springer Journals
Published: Dec 11, 2013
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