Projective Modules Over Central Separable Algebras

Abstract

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.