Let K be a field. The Primitive Element Theorem says that every (finite dimensional) separable extension of K can be generated by 1 element. A folklore theorem says that every central simple K-algebra (e.g., a matrix algebra) can be generated by 2 elements. Can one prove similar bounds on the number of generators for suitable algebras over (commutative) rings? I will discuss some recent progress on this problem based on geometry: both lower and upper bounds may be obtained by studying the space of r-tuples of generators of the algebra.

For example, the R-algebras which naturally generalize the central simple K-algebras of degree n are known as Azumaya algebras of degree n. We show that if R is finitely generated over K and has Krull dimension d, then every such algebra can be generated by 2 + floor(d/(n-1)) elements, and there are examples requiring at least half that many elements to generate.