Finite Rank Torsion Free Abelian Groups and Rings Notation and preliminaries Types and rank—1 groups Examples of indecomposable groups and direct sums Endomorphism rings and decompositions of rank 2 groups Pure subgroups of completely decomposable groups Homogeneous completely decomposable groups and generalizations Completely decomposable groups and generalizations Additive categories quasi isomorphism and near isomorphism Stable range substitution cancellation and exchange properties Subrings of finite dimensional Q algebras Orders in finite dimensional simple Q algebras Maximal orders in finite dimensional simple Q algebras Near isomorphism and genus class Grothendieck groups Additive groups of subrings of finite dimensional Q algebras Q simple and p simple groups.