Editing Direct sum of modules (section)

== Grothendieck group ==
The direct sum gives a collection of objects the structure of a [[Monoid#Commutative_monoid|commutative]] [[monoid]], in that the addition of objects is defined, but not subtraction. In fact, subtraction can be defined, and every commutative monoid can be extended to an [[abelian group]]. This extension is known as the [[Grothendieck group]]. The extension is done by defining equivalence classes of pairs of objects, which allows certain pairs to be treated as inverses. The construction, detailed in the article on the Grothendieck group, is "universal", in that it has the [[universal property]] of being unique, and homomorphic to any other embedding of a commutative monoid in an abelian group.