On Automorphisms and Commutativity in Semiprime Rings

Liau, Pao-Kuei ; Liu, Cheng-Kai

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 584-592

Résumé

Let $R$ be a semiprime ring with center $Z\left( R \right)$ . For $x,\,y\,\in \,R$ , we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$ . If $\sigma $ is a non-identity automorphism of $R$ such that 1 $$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot\right],\,{{x}^{nk}} \right]\,=\,0$$ for all $x\,\in \,R$ , where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$ . In particular, when $R$ is a prime ring, $R$ is commutative.

Détail

DOI : 10.4153/CMB-2011-185-5

Mots-clés : 16N60, 16W20, 16R50, semiprime ring, automorphism, generalized polynomial identity (GPI)