DOI

10.17077/etd.5mkkf904

Document Type

Dissertation

Date of Degree

Spring 2013

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Anderson, Daniel

First Committee Member

Anderson, Dan D.

Second Committee Member

Bleher, Frauke

Third Committee Member

Camillo, Victor

Fourth Committee Member

Jorgensen, Palle

Fifth Committee Member

Krishnamurthy, Muthu

Abstract

Anderson and Frazier defined a generalization of factorization in integral domains called tau-factorization. If D is an integral domain and tau is a symmetric relation on the nonzero nonunits of D, then a tau-factorization of a nonzero nonunit a in D is an expression a = lambda a_1 ... a_n, where lambda is a unit in D, each a_i is a nonzero nonunit in D, and a_i tau a_j for i != j. If tau = D^# x D^#, where D^# denotes the nonzero nonunits of D, then the tau-factorizations are just the usual factorizations, and with other choices of tau we get interesting variants on standard factorization. For example, if we define a tau_d b if and only if (a, b) = D, then the tau_d-factorizations are the comaximal factorizations introduced by McAdam and Swan. Anderson and Frazier defined tau-factorization analogues of many different factorization concepts and properties, and proved a number of theorems either generalizing standard factorization results or the comaximal factorization results of McAdam and Swan. Some of these concepts include tau-UFD's, tau-atomic domains, the tau-ACCP property, tau-BFD's, tau-FFD's, and tau-HFD's. They showed the implications between these concepts and showed how each of the standard variations implied their tau-factorization counterparts (sometimes assuming certain natural constraints on tau). Later, Ortiz-Albino introduced a new concept called Gamma-factorization that generalized tau-factorization. We will summarize the known theory of tau-factorization and Gamma-factorization as well as introduce several new or improved results.

Keywords

commutative algebra, factorization

Pages

vi, 152 pages

Bibliography

Includes bibliographical references (pages 151-152).

Copyright

Copyright 2013 Jason Robert Juett

Included in

Mathematics Commons

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