The class G G of self-small mixed abelian groups of finite rank has recently been the focus of a number of investigations. It is known that, up to quasi-isomorphism, G G is dual to the category of locally free torsion-free abelian groups of finite rank. We utilize the reduced mixed groups in G G as building blocks of a more extensive class ร G G, the smallest class containing the reduced groups in G G that is closed under taking infinite direct sums and direct summands. Our central result is the determination of a complete set of isomorphism invariants for the groups in ร G G. We supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in ร G G. แฎ 1999 Academic Press
1. CANCELLATION AND THE CLASS G G
A ring R has one in the stable range if, whenever the equation r s q 1 1 r s s 1 holds in R, there exists an element s g R such that r q r s is a 2 2 1 2 w x unit of R. In B , Bass proves that if R factored by its Jacobson radical is Artinian, then one is in the stable range of R. Plainly then, finite rings and finite dimensional rational algebras have one in the stable range.
All groups considered will be additive abelian groups and all homomor-ลฝ . phisms will be Z-homomorphisms. For a group G we let E G be the ring ลฝ .