Generators of Central Simple Algebras

LetAbe a finite-dimensional central simple algebra and letkbe a subfield of its centerZ(A). We say thatz1,…,zmgenerateAas a central simple algebra overkifA = S − 1k[z1,…,zm], whereS = k[z1,…,zm] ∩ Z(A)*. In this paper we give a necessary and sufficient condition forAto be generated bymelements as a central simple algebra overk.