On a generalization of a Prüfer-Kaplansky-Procházka theorem. (English) Zbl 0717.13005

In generalization of a sequence of earlier results, it is shown that if \(A\) is a torsion free module over an almost maximal valuation domain, \(R\), then \(A\) is free if and only if \(\tilde R\otimes_ RA\) is an \(\tilde R\)-homogeneous \(\tilde R\)-module and \(A\) belongs to the Baer class \({\mathfrak B}(R)\). Here, \(\tilde R\) is the completion of \(R\) in the ideal topology and \(R\) is \(\tilde R\)-homogeneous if and only if every rank one pure submodule is isomorphic to \(\tilde R\).


13C05 Structure, classification theorems for modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13F30 Valuation rings
20K20 Torsion-free groups, infinite rank
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