• Alok Prasad Rout
• N. Jagannadham

According to W. Keith, 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 K0(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 K0(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.

K-Theory, linear algebra, polynomial ring

"Exploring the beginning of Algebraic K-Theory", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.8, Issue 8, page no.b871-b877, August-2021, Available :http://www.jetir.org/papers/JETIR2108239.pdf

"Exploring the beginning of Algebraic K-Theory", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.8, Issue 8, page no. ppb871-b877, August-2021, Available at : http://www.jetir.org/papers/JETIR2108239.pdf