On the analogue of Corner's finite rank theorem for modules over valuation domains (Q1325091)

In a fundamental paper on endomorphism rings of Abelian groups, \textit{A. L. S. Corner} [Proc. Lond. Math. Soc., III. Ser. 13, 687-710 (1963; Zbl 0116.024)] proved that every countable reduced torsion-free ring is an endomorphism ring of a countable reduced torsion-free Abelian group and moreover if the ring has finite rank \(n\), then the group may be chosen to have rank \(2n\). While the first result has been the starting point for a large body of research, comparatively little attention has been paid to the finite rank result. It is claimed in the recent book by \textit{L. Fuchs} and \textit{L. Salce} [Modules over Valuation Rings (1985; Zbl 0578.13004)] that a similar result holds for modules over valuation domains. The authors show that this is not so by exhibiting a valuation domain \(R_ 0\) such that no finite rank torsion-free \(R_ 0\)-module can have endomorphism algebra \(R_ 0 \times R_ 0\). The construction is based on recent work of \textit{P. Vámos} on decomposition problems for modules [J. Lond. Math. Soc., II. Ser. 41, 10-26 (1990; Zbl 0736.13013)].