# Grothendieck group

## Abelian group extending a commutative monoid / From Wikipedia, the free encyclopedia

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In mathematics, the **Grothendieck group**, or **group of differences**,[1] of a commutative monoid *M* is a certain abelian group. This abelian group is constructed from *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.