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

### ON (ANTI-)MULTIPLICATIVE GENERALIZED DERIVATIONS

### 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)c*^{2},
*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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