Let R be a ring. We say x ∈ R is clean if x = e + u where u is a unit and e is an idempotent (e2 = e). R is clean if every element of R is clean. I will give the motivation for clean rings, which comes from Fitting's Lemma for Vector Spaces. This leads into the ABCD lemma, which is the foundation of a paper by Camillo, Khurana, Lam, Nicholson and Zhou. Semi-perfect rings are a well known type of ring. I will show a relationship that occurs between clean rings and semi-perfect rings which will allow me to utilize what is known already about semi-perfect rings. It is also important to note that I will be using the Fundamental Theorem of Torsion-free Modules over Principal Ideal Domains to work with finite dimensional vector spaces. These finite dimensional vector spaces are in fact strongly clean, which simply means they are clean and the idempotent and unit commute. This additionally means that since L = e + u, Le = eL. Several types of rings are clean, including a weaker version of commutative Von Neumann regular rings, Duo Von Neumann regular, which I have proved. The goal of my research is to find out how many ways to write matrices or other ring elements as sums of units and idempotents. To do this, I have come up with a method that is self contained, drawing from but not requiring the entire literature of Nicholson. We also examine sets other than idempotents such as upper-triangular and row reduced and examine the possibility or exclusion that an element may be represented as the sum of a upper-triangular (resp. row reduced) element and a unit. These and other element properties highlight some of the complexity of examining an additive property when the underlying properties are multiplicative.
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-5886 |
| Date | 01 July 2015 |
| Creators | Borchers, Brian Edward |
| Contributors | Camillo, Victor |
| 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 2015 Brian Edward Borchers |
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