Model theory of abelian groups

Abstract

This thesis is mainly concerned with the following sort of question. Let A and C be Abelian groups with certain model-theoretic properties. What can be deduced about the model-theoretic properties of F(A,C) where F is some operation on the class of Abelian groups?

Since we never consider any other sort of group, we use the term group to mean Abelian group throughout. We give the following results: in chapter 1, a characterisation of groups A such that for any group A', A ≡ A' if and only if T(A) ≡ T(A') and A/T(A) ≡ A'/T(A'): in chapter 2 we shew that the above characterisation also characterises groups A and C such that for any groups A' and C', A ≡ A' and C ≡ C' implies A ⊗ C ≡ A' ⊗ C': in chapter 3 we shew that for any groups A, A', C, C' A ≡ _kw A' and C ≡ _kw C' implies Tor(A,C) ≡ _kw Tor(A', C'): in chapter 4 we obtain some results about groups without elements of infinite height: in chapters 5 and 6 we extend our investigations to Hom and Ext.

Finally in the appendix, we shew how to extend most of these results to modules over Dedekind domains.