In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa1 · · · an where λ is in U (D) and ai is τ-related to aj, denoted ai τ aj, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization.
Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b)∗ = D. We then write a τmax b (resp. a τv b, a τ[ ] b, or a τ∗ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-1770 |
| Date | 01 May 2010 |
| Creators | Reinkoester, Jeremiah N |
| Contributors | Anderson, Daniel D., 1948- |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright 2010 Jeremiah N Reinkoester |
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