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 : Mi → M 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:
“The responsible business men of this country put their shoulders to the wheel. It is in response to this universal demand that we are founding today, All-American Airways.”
—John Dos Passos (1896–1970)
“It is better to write of laughter than of tears, for laughter is the property of man.”
—François Rabelais (1494–1553)