On rings with distinguished ideals and their modules.




Buckner, Joshua.

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Let S be an integral domain, R an S algebra, and F a family of left ideals of R. Define End(R, F) = {φ ∈ End(R+) : φ(X ) ⊆ X for all X ∈ F }. 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 ⊆ QR and End(M+) = R. This motivates the following definitions: Call R 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 R is a Zassenhaus ring then R has a family F of left ideals such that End(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 ⊆ QR and End(M+) = R.


Includes bibliographical references (p. 54-55).