Exploring the Beginnings of Algebraic K-Theory

According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K 0 ( R ) as the free abelian group whose basis are the finitely generated projective R -modules with the added relation P ⊕ Q = P + Q . The purpose of this thesis is to study simple settings of the K-theory for rings and to provide a sequence of examples of rings where the associated K -groups K 0 ( R ) get progressively more complicated. We start with R being a field or a principle ideal domain and end with R being a polynomial ring on two variables over a non-commutative division ring.