On rings with distinguished ideals and their modules.

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Doctoral

Grantor

Baylor University.

Year dc:date.issued

2007

Author dc:creator

• Buckner, Joshua.

Advisor dc:contributor.advisor

• Dugas, Manfred.

• Modules (Algebra)
• Abelian group theory.

Statement dc:rights

• Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission.

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/2104/5025

OAI identifier oai:identifier

oai:baylor-ir.tdl.org:2104/5025