On the quasi-splitting of exact sequences

Abstract

Perhaps the most significant difference between the theories of torsion and torsion-free abelian groups is the large number of splitting results in the former, and the lack of such in the latter. For instance, Szele’s theorem [F, Vol. 1, Proposition 27.11 has no obvious equivalent in the theory of torsion-free abelian groups, although it is one of the widely used results in the discussion of torsion and mixed groups: Let B g QJI Z/p’7 be a subgroup of an abelian group G. Then B is a direct summand of G if and only if B np”G = 0. The author was able to extend Szele’s theorem to valuated p-groups in [A2, Satz 6.81 using methods similar to those which proved useful in the discussion of Baer’s lemma in [AL], [A3], [A4]. It is the goal of this paper to prove a result which can be considered the equivalent of Szele’s theorem in the quasi-category of torsion-free abelian groups. Before we can describe the result, it is necessary to introduce some notation. We consider an abelian group A. The class of A-projective groups of finite A-rank is obtained by closing {A} with respect to direct summands and finite direct sums. If G is an abelian group, then the A-radical of G, which is denoted by R,(G), is R,(G) = n {ker 41 #E Hom(G, A)). Finally, consider modules B and C over a ring R. A homomorphism ~1: B + C quasi-splits if there are a non-zero integer n and a map /3~Hom,(C, B) with /?cr=n.id,, where id, denotes the identity map on B. If the additive group of B is torsion-free, then a quasi-splitting homomorphism is one-to-one [Jl, 52, Re]. We say that an abelian group A has the radical-splitting property if every exact sequence 0 + PA G of torsion-free abelian groups, in which P is a quasi-summand of an A-projective group of finite A-rank and u(P) n R,(G) = 0, quasi-splits. Szele’s original result guarantees the splitting of the sequence in the case in which A z Z/p’7 for some n < w. Moreover, if A z Z/p”Z, then R,(G) = p”G. Theorem 3.1 of this paper establishes that every strongly indecomposable group of finite rank has the