Finite representation type and direct-sum cancellation

Consider the notion of finite representation type (FRT for short): An integral domain R has FRT if there are only finitely many isomorphism classes of indecomposable finitely generated torsion-free R-modules. Now specialize: Let R be of the form D + cO where D is a principal ideal domain whose residue fields are finite, c is an element of D is a nonzero nonunit, and O is the ring of integers of some finite separable field extension of the quotient field of D. If the D-rank of R is at least four then R does not have FRT. In this case we show that cancellation of finitely generated torsion-free R-modules is valid if and only if every unit of O/cO is liftable to a unit of O. We also give a complete analysis of cancellation for some rings of the form D + cO having FRT. We include some examples which illustrate the difficult cubic case. (C) 2004 Elsevier Inc. All rights reserved.