Cotilting and a hierarchy of almost cotorsion groups

We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V = L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groups-called almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group. (C) 2000 Academic Press.