Grothendieck Group
The direct sum gives a collection of objects the structure of a 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 an abelian monoid in an abelian group.
Read more about this topic: Direct Sum Of Modules
Famous quotes containing the word group:
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)