Volume 36 • Number 4 • 2013   • Generalization of Posner's Theorems Ahmed A. M. Kamal and Khalid H. Al-Shaalan} Abstract. In this paper we generalize Posner's first theorem to a 3-prime near-ring with a $(\sigma ,\tau )$-derivation. We prove that a prime ring with a non-zero $(\sigma ,\tau )$-derivation is commutative if $\sigma (x)d(x)=d(x)\tau (x)$ for all $x\in U$ where $U$ is a suitable subset of $R$%. Also, we generalize Posner's second theorem completely to a prime ring with a $(\sigma ,\sigma )$-derivation and partially to a prime ring with a $% (\sigma ,\tau )$-derivation. 2010 Mathematics Subject Classification: 16W25, 16Y30 Full text: PDF