A constructive theory of minimal zero-dimensional extensions

Abstract

Chiorescu characterized the minimal zero-dimensional extensions of certain one-dimensional rings in terms of families of ideals indexed by prime ideals. In this paper we give a constructive development of these extensions, which, to achieve maximum generality, must necessarily avoid dependence on prime ideals. This forces us to develop a purely arithmetic theory. Along the way we get a characterization, in terms of the lattice of radicals of finitely generated ideals, of when a ring with primary zero-ideal has dimension at most one.