Glasnik Matematicki, Vol. 47, No. 1 (2012), 105-118.

ON (ANTI-)MULTIPLICATIVE GENERALIZED DERIVATIONS

Daniel Eremita and Dijana Ilišević

Department of Mathematics, University of Zagreb, Bijenička 30, P.O.Box 335, 10002 Zagreb, Croatia
e-mail: ilisevic@math.hr

Abstract.   Let R be a semiprime ring and let F, f : R → R be (not necessarily additive) maps satisfying F(xy)=F(x)y+xf(y) for all x,y R. Suppose that there are integers m and n such that F(uv)=mF(u)F(v)+nF(v)F(u) for all u, v in some nonzero ideal I of R. Under some mild assumptions on R, we prove that there exists c C(I^⊥⊥) such that c=(m+n)c², nc[I^⊥⊥, I^⊥⊥]=0 and F(x)=cx for all x I^⊥⊥. The main result is then applied to the case when F is multiplicative or anti-multiplicative on I.

2010 Mathematics Subject Classification.   16U99, 16N60, 39B52, 47B47.

Key words and phrases.   Additivity, ring, semiprime ring, prime ring, derivation, generalized derivation, homomorphism, anti-homomorphism.

DOI: 10.3336/gm.47.1.08

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