Radicals in Abelian Groups

Abstract

This chapter discusses radicals in abelian groups. The set theoretical and model theoretical methods are powerful tools for investigating abelian groups. The author intends to use the combinatorial methods in radical theory and to show that many of the recent results in abelian groups can be applied to settle open questions about radicals. An abelian group G is called cotorsion-free if G does not have any nontrivial cotorsion subgroups or equivalently G is reduced, torsion-free and Jp is not isomorphic to a subgroup of G for all primes. The torsion theory is not singly cogenerated but singly generated. If v is the Chase radical, then the torsion theory is singly generated but not singly cogenerated.