Absolutely pure modules

Introduction. In this paper several properties of absolutely pure modules are given. It is shown that absolutely pure and injective are equivalent properties for modules over Dedekind rings. However, it is proved that absolutely pure and injective are not equivalent properties for modules over rings which are not Noetherian. That every module has a maximal absolutely pure submodule is also established. A sufficient condition for the uniqueness of a maximal absolutely pure submodule is also given. This paper constitutes a portion of the author's doctoral dissertation written at the University of South Carolina where he held a Cooperative Graduate Fellowship. The writer is indebted to Professor Edgar Enochs who suggested this topic and directed its development while providing sufficient inspiration and assistance and, most of all, exhibiting infinite patience. In this paper all rings will have a unit and all modules will be unitary. A will always denote a ring. We agree that if E' is a submodule of E and v. E'—>E is the canonical injection then the map 1 ®v: F®E' —>P £ will be called the canonical map where 1: F-^F is the identity map of P. If the canonical map is an injection for all P, then E' is said to be a pure submodule of P. Observe that if E' is a pure submodule of E then aEC\E' =aE' for all nonzero aEA by examining the diagram