GENERATORS OF MAXIMAL ORDERS

Abstract Let R be the ring of algebraic integers in a number field K and let Λ be a maximal order in a finite dimensional semisimple K -algebra B . Building on our previous work [3] , we compute the smallest number of algebra generators of Λ considered as an R -algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.