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 axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)
“Let the amelioration in our laws of property proceed from the concession of the rich, not from the grasping of the poor. Let us understand that the equitable rule is, that no one should take more than his share, let him be ever so rich.”
—Ralph Waldo Emerson (1803–1882)