RINGS WITH TOPOLOGIES INDUCED BY SPACES OF FUNCTIONS

ABSTRACT. By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions, we prove that if an ideal is closed then every prime ideal associated to it is closed (thus answering a question raised in [5]). The converse is also true if we assume that a topo-logical version of the Nullstellensatz holds, and we prove such a result for the ring of polynomials in two variables endowed with the topology induced by the Hardy space. The topological completion of the ring is a module, and we show the existence of a one-to-one correspondence between closed ideals of finite codimension and closed submodules of finite codimension which preserves primary decompositions. At the end we consider the case of the Hardy space on the polydisc, and of Bergman spaces on the unit ball and Reinhardt domains. 0. Introduction. In the development of multivariable operator theory there appears to exist a fundamental connection between the study of invariant subspaces fo