Uniquely clean elements, optimal sets of units and counting minimal sets of units

Abstract

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 (e = 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 [3]. 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. [3] It is also important to note that I will be using the Fundamental Theorem of Torsion-free Modules over Principal Ideal Domains [6] 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 for a linear transformation L, Le = eL Several types of rings are clean, including a weaker version of commutative von Neumann regular rings, duo von Neumann regular. [7] 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 [8]. We also examine sets other than idempotents such as upper-triangular matrices and row reduced matrices and examine the possibility or exclusion that an element