Direct Sum of Modules - Universal Property

Universal Property

In the language of category theory, the direct sum is a coproduct and hence a colimit in the category of left R-modules, which means that it is characterized by the following universal property. For every i in I, consider the natural embedding

which sends the elements of Mi to those functions which are zero for all arguments but i. If fi : MiM are arbitrary R-linear maps for every i, then there exists precisely one R-linear map

such that f o ji = fi for all i.

Dually, the direct product is the product.

Read more about this topic:  Direct Sum Of Modules

Famous quotes containing the words universal and/or property:

    Let us build altars to the Blessed Unity which holds nature and souls in perfect solution, and compels every atom to serve an universal end.
    Ralph Waldo Emerson (1803–1882)

    The diversity in the faculties of men, from which the rights of property originate, is not less an insuperable obstacle to a uniformity of interests. The protection of these faculties is the first object of government.
    James Madison (1751–1836)