#### Publication Date

5-2021

#### Date of Final Oral Examination (Defense)

3-3-2021

#### Type of Culminating Activity

Thesis

#### Degree Title

Master of Science in Mathematics

#### Department

Mathematics

#### Major Advisor

Jens Harlander, Ph.D.

#### Advisor

Zachariah Teitler, Ph.D.

#### Advisor

Uwe Kaiser, Ph.D.

#### Abstract

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.

#### Recommended Citation

Schott, Sarah, "Exploring the Beginnings of Algebraic K-Theory" (2021). *Boise State University Theses and Dissertations*. 1798.

https://scholarworks.boisestate.edu/td/1798