Grothendieck Rings And Grothendieck Moudles Of Ableian Tensor Category

The Grothendieck group plays an important role in many fields,such as algebra, number theory and algebraic geometry.It may have a ring structure induced by tensor product and is called a Grothendieck ring.However in many interesting cases,we need to define both Grothendieck rings and Grothendieck modules.In this paper we define Grothendieck rings and Grothendieck modules for general abelian tensor category.We define Tor functors and give some basic properties of Grothendieck rings and Grothendieck modules.