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:
“We can most safely achieve truly universal tolerance when we respect that which is characteristic in the individual and in nations, clinging, though, to the conviction that the truly meritorious is unique by belonging to all of mankind.”
—Johann Wolfgang Von Goethe (1749–1832)
“You and I ... are convinced of the fact that if our Government in Washington and in a majority of the States should revert to the control of those who frankly put property ahead of human beings instead of working for human beings under a system of government which recognizes property, the nation as a whole would again be in a bad situation.”
—Franklin D. Roosevelt (1882–1945)