On rings with distinguished ideals and their modules

Let S be an integral domain, RS an S algebra, and F a family of left ideals of R. Define End S(R, F ) = {lcub}ϕ ∈ EndS( R+) : ϕ(X) ⊆ X for all X ∈ F {rcub}. In 1967, H. Zassenhaus proved that if R is a ring such that R+ is free of finite rank, then there is a left R module M such that R ⊆ M ⊆ Q R and EndZ (M) = R. This motivates the following definitions: Call RZ a Zassenhaus ring with module M if the conclusion of Zassenhaus' result holds for the ring R and module M. It is easy to see that if RZ is a Zassenhaus ring then R has a family F of left ideals such that EndZ (R, F ) = R. (If F has this property, then call F a Zassenhaus family (of left ideals) of the ring R.) While the converse doesn't hold in general, this dissertation examines examples of rings R for which the converse does hold, i.e. R has a Zassenhaus family F of left ideals that can be used to construct a left R module M such that R ⊆ M ⊆ Q R and EndZ (M) = R.