Definable compactness and definable subgroups of o-minimal groups

The paper introduces the notion of definable compactness and within the con text of o-minimal structures proves several topological properties of defin ably compact spaces. In particular a definable set in an o-minimal structur e is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the s tudy of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not de finably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings withou t zero divisors that are definable in o-minimal structures. The paper concl udes with several examples illustrating some limitations on extending the t heorem.