### Daniel Eremita and Dijana Ilišević

Department of Mathematics and Computer Science, FNM, University of Maribor, 2000 Maribor, Slovenia
e-mail: daniel.eremita@uni-mb.si

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.

Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.08

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