ON THE BAER-KAPLANSKY THEOREM FOR TORSION FREE GROUPS WITH QUADRATIC SPLITTING FIELDS

Abstract

The connection between a structure of abelian group and a structure of endomorphism ring is a classic question in abelian group theory. In particular, Baer and Kaplansky proved that this connection is very strong for torsion groups: two abelian torsion groups are isomorphic if and only if their endomorphism ring are isomorphic. In more general cases for torsion-free and mixed abelian groups the Baer-Kaplansky theorem is fails. This paper deals with a class of p-local torsion-free abelian of finite rank. Let K be a field such that Q ⊂ K ⊂ Qbp and let R = K ∩ Zbp, where Zbp is the ring of p-adic integers, Qbp is the field of p-adic numbers, Q is the field of rational numbers. We say that K is a splitting field (R is a splitting ring) for a p-local torsion-free reduced group A or that group A is K-decomposable group if A ⊗Zp R is the direct sum of a divisible R-modules and a free R-modules. Torsion-free p-local abelian groups of finite rank with quadratic splitting field K are characterized. As an application it is proved that K-decomposable plocal torsion free abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.