Localization of stable homotopy rings

V. P. Snaith has shown that a number of interesting homology theories (e.g. K -theory, complex cobordism) arise when one localizes the stable homotopy type of certain H -spaces ( BU (1), BU , etc.). In this paper we sketch a construction which, given an H-space X and , produces a ring spectrum IΣ −n X which lies between X and its localization with respect to b , and in many cases has a tractable structure. We apply this when X is an Eilenberg-MacLane space K ( ℤ/, n ) and is a generator ( p = 0 or a prime), and get complete results on the localizations in this case. The results confirm one's suspicions that localization is a fairly drastic manoeuvre, cutting down the large and complicated ring to something more or less trivial, except when p = 0 and n = 2, in which case the rich geometry of K (ℤ, 2) = BU (1) gives complex K -theory.