Valuation Domains with Superdecomposable Pure Injective Modules

Abstract

The first important extension of the theory of pure injective (i.e. algebraically compact) abelian groups, developed in the ’50’s by Kaplansky [K], Łoś [L], Maranda [Mar] and others, to modules over arbitrary associative rings with 1, was given by Warfield in [W1], In section 6 of that paper, Warfield considered the special case of pure injective modules over a Prüfer ring R, and showed that such a module M decomposes as M = E ⊕ N, where E is injective and the first Ulm submodule N* = ∩{rN | 0 ≠ r ∈ R} of N is vanishing. If R is h-local (see [Mat]), then the injective summand E can be completely classified (see [W2]), and the summand N is a direct product N = ∏ N m of pure injective modules over the localizations R m (where m ranges over all maximal ideals of R), which are valuation domains. The last step in section 6 of [W1], in trying to determine the structure of general pure injective modules over valuation domains, is Theorem 5, which furnishes a complete classification in the torsionfree case.