ABSTRACT
In all cases the categories may be considered with near-homomorphisms as maps. There is a rather complicated definition of near-homomorphism in [Lad75] valid for all torsion-free groups of finite rank, that has not been used widely. There are more palatable versions in [Arn82]. If a finite set of primes P is fixed and one considers all those almost completely decomposable groups whose regulating indices have only prime factors in P , then one may take as morphism groups the localizations Hom(X, Y )p = Zp®Hom(X, Y). It has been shown that isomorphism in this category coincides with near-isomorphism (Theorem 9.2.10). This idea appears in [Sch95] and [Ric94]. These near-homomorphism categories carry an automatic concept of direct summand and the connection with the various ad hoc definitions of near-summand needs to be clarified ([Arn82], [Lad95]).
