Superdecomposable pure injective modules over commutative valuation rings
A pure-injective module is continuous (superdecomposable) if it has no nonzero indecomposable direct summands. \textit{M. Ziegler} [Ann. Pure Appl. Logic 26, 149-213 (1984; Zbl 0593.16019)] claimed that there are no continuous pure-injectives over a commutative valuation domain, a claim repeated in the book of \textit{M. Prest} [``Model theory and modules'', Lond. Math. Soc. Lect. Note Ser. 130 (1988; Zbl 0634.03025)]. On the other hand \textit{L. Fuchs} and \textit{L. Salce} [``Modules over valuation domains'', Lect. Notes Pure Appl. Math. 97 (1985; Zbl 0578.13004)] refer to the existence of such modules. In this paper the author resolves the contradiction by giving a characterization of exactly when a commutative valuation ring \(R\) has continuous pure-injectives. The tools used will be of some interest: there are useful results on the lattice of pp formulas (in one variable) over a commutative valuation ring, and useful results describing the pp-1-types. The characterization then follows, with the corollary that for a commutative valuation domain \(R\) with value group \(\Gamma\), there is a continuous pure-injective module over \(R\) if and only if there is an embedding of the rational order in \(\Gamma\). -- Finally there is an analysis of the error in Ziegler's original proof: Ziegler claims that the ``width'' of the lattice of pp-1-formulas over a commutative valuation domain \(R\) is not greater than 2. This claim is shown to be false even for the simple case of the localization of the integers at a prime \(p\).
scientific article; zbMATH DE number 536683