Glasnik Matematicki, Vol. 47, No. 2 (2012), 307324.
COMPOSITION OF GENERALIZED DERIVATIONS AS A LIE DERIVATION
Vincenzo De Filippis and Giovanni Scudo
Di.S.I.A., University of Messina, 98166 Messina, Italy
email: defilippis@unime.it
Department of Mathematics, University of Messina, 98166 Messina, Italy
email: gscudo@unime.it
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 nonzero 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}.
2010 Mathematics Subject Classification.
16N60, 16W25.
Key words and phrases. Prime rings, differential identities, generalized derivations.
Full text (PDF) (free access)
DOI: 10.3336/gm.47.2.07
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