Date of Degree
PhD (Doctor of Philosophy)
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.
A matrix is a rectangular array of numbers. Matrices occur commonly in all areas of pure and applied mathematics. Of the properties that matrices may have, being invertible or idempotent are among the most important. It is known that every square matrix is a sum of an invertible matrix and an idempotent matrix, this is called Fittings Lemma.
In this paragraph we abuse language a bit to communicate informally. This is a thesis in pure mathematics written in an applied mathematical spirit. It is, broadly thought of, an exercise in computing efficiency. We say a specific set of idempotents E generates all matrices if every matrix is a sum of an element in E and some invertible matrix. We are specifically interested in the so called 0; 1 diagonal idempotents, (i.e. square matrices whose entries on the diagonal are 0 or 1 and entries off the diagonal are 0) the most important set in all of matrix theory. We ask the question, what is the smallest set of units required to generate all matrices with E? This number is hard to compute. For 2 by 2 matrices with entries in integers mod 3, the smallest number of units required is 32.
Specifically this thesis both computes and estimates. Our estimates actually yield a very interesting number in a limiting case and suggest quite an engaging but difficult conjecture. In our specific cases we obtain concrete numbers but the computations are quite technical. We also obtain a result for upper triangular matrices. You can think of an upper triangular matrix as a table in which all the entries in the southwest corner are zero. Finally we study idempotents and so called row reduced matrices. These matrices are the foundation stone of the procedure used to solve systems of linear equations. In the 3 by 3 matrix case we do extensive computations to determine which matrices are the sum of a matrix from our distinguished set of diagonal idempotents and a row reduced matrix.
publicabstract, Algebra, clean, idempotents, matrices, units
Copyright 2015 Brian Edward Borchers