Grothendieck group

Summary

The Grothendieck group of a commutative monoid is the abelian group obtained by adding formal differences, in the universal way. It turns the natural numbers into the integers. Applied to isomorphism classes of finitely generated projective modules under direct sum, it gives the group $K_0$ of a ring, the start of algebraic K-theory. The same idea applies to vector bundles and to abelian categories.