Torsion-free modules of finite balanced-projective dimension over valuation domains

Let R be a valuation domain. Several characterizations are obtained of tors ion-free R-modules having finite balanced-projective dimension (bpd) by usi ng the new concept of an n-balanced submodule and a special pull-back diagr am. Noteworthy among the Various results are: For torsion-free R-modules M, bpd(M) less than or equal to pd(M), where pd(M) denotes the projective dim ension of M. If M is a torsion-free R-module which is not balanced-projecti ve, then bpd(M)= pd(M/B) for any basic submodule of M. (C) 2001 Elsevier Sc ience B.V. All rights reserved. MSC: Primary 13F30; 13D05; 16E10; secondary 16D40; 18G20.