×

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 \(\tilde R\).

MSC:

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
PDFBibTeX XMLCite
Full Text: EuDML