TORSION-FREE EXTENSIONS OF PROJECTIVE MODULES BY TORSION MODULES

We consider a generalization of a problem raised by P. Griffith on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F & SIM;= T, then M is projective. It is shown that for abelian groups whose cardinality is not cofinal with & omega; this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy. The problem for valuation domains is also discussed, with results similar to the case of abelian groups.