were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent findings. Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are: Tensor product A.L.S. Corner's results on countable torsion-free groups Shelah's work to remove cardinality restrictions Burnside ring Relation to other mathematical topics Many large abelian groups possess a natural topology, which turns them into topological groups. The collection of all abelian groups, together with the homomorphisms between them, forms the category , the prototype of an abelian category. proved that the first-order theory of abelian groups, unlike its non-abelian counterpart, is decidable. Most algebraic structures other than Boolean algebras are undecidable. There are still many areas of current research: Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood; There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups; While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature. Many mild extensions of the first-order theory of abelian groups are known to be undecidable. Finite abelian groups remain a topic of research in computational group theory. Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is: Undecidable in ZFC (Zermelo–Fraenkel axioms), the conventional axiomatic set theory from which nearly all of present-day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC; Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom; Positively answered if ZFC is augmented with the axiom