Original scientific paper

https://doi.org/10.3336/gm.47.2.07

Composition of generalized derivations as a Lie derivation

Vincenzo De Filippis orcid.org/0000-0002-3213-1254 ; Di.S.I.A., University of Messina, 98166 Messina, Italy
Giovanni Scudo ; Department of Mathematics, University of Messina, 98166 Messina, Italy

Abstract

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds:

there exist α C and a U such that F(x)=[a,x] and G(x)=α x, for all x R;
G is an usual derivation of R and there exists α C such that F(x)=α x, for all x R;
there exist α, β C and a derivation h of R such that F(x)=α x+h(x), G(x)=β x, for all x R, and α β+h(β)=0. Moreover in this case h is not an inner derivation of R;
there exist a', c' U such that F(x)=a'x, G(x)=c'x, for all x R, with a'c'=0;
there exist b', q' U such that F(x)=xb', G(x)=xq', for all x R, with q'b'=0;
there exist c', q' U, η, γ C such that F(x)=η (xq'-c'x)+γ x, G(x)=c'x+xq', for all x R, with γ c'-η c'2=-γ q'-η q'2.

Keywords

Prime rings; differential identities; generalized derivations

Hrčak ID:

93946

URI

https://hrcak.srce.hr/93946

Publication date:

19.12.2012.

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