Generators of maximal orders. (Q2255059)

In 1974, Pleasants has determined the number of generators of the ring of integers of a number field \(K\) in terms of local invariants \(m_P\) of the primes \(P\) in \(K\). In the paper under review, these results are extended to maximal orders in a module-finite separable algebra. A particularly nice theorem is obtained for maximal orders \(\Lambda\) over a Dedekind domain \(R\) with perfect infinite residue class fields. It is proved that under this assumption, the \(R\)-algebra \(\Lambda\) can be generated by three elements. It remains open whether the number of generators might even be two. Remarks. The theorem can be viewed as a variant of the classical fact that smooth projective curves can always be represented in the projective three-space, but not in the plane if the secant variety fills the whole three-space. Of course, orders over a Dedekind domain are affine, but the exceptional rôle of infinite primes has to be noticed, as well as the fact that they represent non-commutative curves.