Periodica Mathematica Hungarica Vol. 46 (1), 2003, pp. 33–40 ON THE GLOBAL BALANCED-PROJECTIVE DIMENSION OF VALUATION DOMAINS L. Fuchs (New Orleans) and K. M. Rangaswamy (Colorado Springs) [Communicated by: M´aria B. Szendrei] Abstract Assuming the General Continuum Hypothesis, the global balanced-projective dimension of a valuation domain is determined. We show that it is always equal to the supremum of the projective dimensions of torsion-free modules. Let R be any commutative domain with 1, and Q its field of quotients. An exact sequence 0 → A α −→ B β −→ C → 0 of torsion-free R-modules is called balanced if for every rank 1 torsion-free R-module J , the induced map Hom(1 J ,β): Hom R (J, B) → Hom R (J, C) is surjective, i.e., J has the projective property relative to the given exact sequence. The class of balanced-exact sequences form a so-called proper class, and conse- quently, it gives rise to derived functors Bext n R for integers n ≥ 1. The torsion-free balanced-projectives over arbitrary domains are known to be summands of completely decomposable torsion-free R-modules (see Fuchs–Salce [5]); they are themselves completely decomposable in some cases, e.g., for the ring of integers and for valuation domains. There are enough balanced-projectives: every torsion-free R-module M can be embedded in a balanced-exact sequence 0 → N → C → M → 0 with C completely decomposable. Hence the natural question arises to find the balanced projective dimensions (abbreviated: bpd) of torsion-free R- modules, in particular, their supremum, the global balanced-projective dimension of R, gl.bpd R. We emphasize that we restrict the discussion to torsion-free modules, since the torsion case raises questions which would require the development of machinery not available at this time. In case R = Z, we have gl.bpd Z = ∞. In fact, for every non-negative integer n, there are even finite rank torsion-free abelian groups whose balanced projective dimensions are n (cf. Arnold-Vinsonhaler [1]). On the other hand, if R Mathematics subject classification number: 13D05, 13F30. Key words and phrases: valuation domain, balanced submodule, balanced-projective dimension, basic submodule. 0031-5303/03/$20.00 Akad´ emiaiKiad´o,Budapest c  Akad´ emiaiKiad´o,Budapest Kluwer Academic Publishers, Dordrecht