Rings with nondiscrete ideal topologies

Abstract

0. Introduction. A proper topology on a ring A (all rings are supposed commutative, with unit) is simply a nondiscrete Hausdorff ring topology on A. A proper topology will be called an ideal topology if there is a neighborhood basis (equivalently, subbasis) for 0 consisting of ideals. We shall obtain a satisfactory ring-theoretic characterization of those rings which have ideal topologies. In the Noetherian case we get the result that a Noetherian ring has no ideal topology if it is Artin. We also study what happens in case of localization and the taking of direct products. Recently, J. 0. Kiltinen has shown [1, p. 69 ] and [2 ] that every infinite field has a proper topology. Using this fact and our results on ideal topologies we show that every infinite Noetherian ring has a proper topology. We note that most of the ring topologies defined in commutative algebra are ideal topologies, including the usual valuation topologies.