A note on derivations in semiprime ringsVukman, Joso (Avtor) Kosi-Ulbl, Irena (Avtor) mathematicsassociative rings an algebrasderivationssemiprime ringsWe prove in this note the following result. Let ▫$n>1$▫ be an integer and let ▫$R$▫ be an ▫$n!$▫-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping ▫$D : R \to R$▫ such that ▫$D(x^n)=\Sigma_{j^n}=1^{x^{n-j}}D(x)x^{j-1}$▫ is fulfilled for all ▫$ x \in R$▫. In this case, ▫$D$▫ is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, ▫$R$▫ will represent an associative ring with center ▫$Z(R)$▫. Given an integer ▫$n > 1$▫, a ring ▫$R$▫ is said to be ▫$n$▫-torsion-free if for ▫$x \in R$▫, ▫$nx=0$▫ implies that ▫$x=0$▫. Recall that a ring ▫$R$▫ is prime if for ▫$ a,b \in R$▫, ▫$aRb=(0)$▫ implies that either ▫$a=0$▫ or ▫$b=0$▫, and is semiprime in case ▫$aRa=(0)$▫ implies that ▫$a=0$▫. An additive mapping ▫$D:R \to R$▫ is called a derivation if ▫$D(xy)=D(x)y+xD(y)$▫ holds for all pairs ▫$x,y \in R$▫ and is called a Jordan derivation in case ▫$D(x^2)=D(x)x+xD(x)$▫ is fulfilled for all ▫$x \in R$▫. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to ▫$2$▫-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to [2, 7, 8, 9, 10].20052017-06-14 11:22:07Članek v reviji66189sl