Grothendieck Groups of Rings

Algebraic K-theory extends many methods of linear algebra from a number field to a ring R. It employs a set of Abelian groups, denoted \(K_0(R), K_1(R), K_2(R)\) , and so on, using highly sophisticated arithmetic techniques and profound algebraic concepts–hence the term “algebraic,” though its origins lie in practical topology. In this chapter, we will focus solely on introducing the most fundamental of these groups–the Grothendieck group \(K_0(R)\) of a ring R.