Algebraic Properties of Ore Extensions and Their Commutative Subrings

Sammanfattning: This thesis deals with a class of rings known as Ore extensions. An Ore extension can be described as a noncommutative ring of polynomials in one variable. A special focus of the thesis is the study of commuting elements in Ore extensions. In the first two papers in the thesis we prove that commuting elements of Ore extensions are in many cases algebraically dependent. In doing this we extend a classical result for the ordinary ring of polynomials. We also show how to compute the polynomial that annihilates a pair of commuting elements by a construction that generalizes the classical resultant. In the third paper we deal with the simplicity of Ore extension, and give a number of necessary and sufficient conditions. The fourth and sixth paper return to the study of commuting elements and show that the centralizer of an element of an Ore extension is commutative in certain cases, as well as various other properties. In the fifth paper we show that the construction of Ore extensions really gives an associative ring.