Some closure properties for torsion classes of abelian groups

Abstract

In an earlier paper (B. J. Gardner, Pacific J. Math., 33 (1970), 109-116) the torsion classes of abelian groups which are closed under pure subgroups were characterized, and §§ 3-6 of the present paper are devoted to generalizations of results appearing there. If & is a homomorphically closed class of objects in an abelian category, a subobject A of an object B is called ^-pure if it is a direct summand of every intermediate subobject X for which XIA e . (This terminology is due to C. P. Walker). In particular, ^ may be a torsion class. The following question is investigated: If and Ήf are torsion classes of abelian groups, when is closed under ^f-pure subgroups? Although ordinary purity is not ^/-purity for any torsion class %f, a torsion class ^7~~ is closed under pure subgroups if and only if it is closed under ^Hrpure subgroups, where ^l is the class of all torsion groups. In §5, for an arbitrary torsion theory (*g/, ^ ) a rank function (^/-rank) is defined for nonzero groups in ^ . It is shown that every torsion class closed under ^/-pure subgroups is determined by its intersection with ^ and the groups of ^/-rank 1 it contains. When ^ = , the groups with ^ rank 1 are the rational groups, so the earlier results for ordinary purity suggest that in general some refinement of the representation should he possible. A further special case of the general problem is also solved: Let X and Y be rational groups, T(X), T(Y) the smallest torsion classes containing them. If X is a subring of the rationale then T(X) is always closed under T(Γ)-pure subgroups; if not, the condition is satisfied if and only if X has a greater type than Y. § 7 is devoted to proving the following result: A torsion class is closed under countable direct products, i.e. direct products of countable sets of groups, if and only if it is determined by torsion-free groups.