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:
“If only nature is real and if, in nature, only desire and destruction are legitimate, then, in that all humanity does not suffice to assuage the thirst for blood, the path of destruction must lead to universal annihilation.”
—Albert Camus (1913–1960)
“Let’s call something a rigid designator if in every possible world it designates the same object, a non-rigid or accidental designator if that is not the case. Of course we don’t require that the objects exist in all possible worlds.... When we think of a property as essential to an object we usually mean that it is true of that object in any case where it would have existed. A rigid designator of a necessary existent can be called strongly rigid.”
—Saul Kripke (b. 1940)