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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 bytensor product and is called a Grothendieck ring. However in many interestingcases, we need to define both Grothendieck rings and Grothendieck modules. Inthis paper we define Grothendieck rings and Grothendieck modules for generalabelian tensor category. We define Tor functors and give some basic properties ofGrothendieck rings and Grothendieck modules.