Editing Direct sum of modules (section)

== Universal property ==
In the language of [[category theory]], the direct sum is a [[coproduct]] and hence a [[limit (category theory)|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''

:<math>j_i : M_i \rightarrow \bigoplus_{i \in I} M_i</math>

which sends the elements of ''M''<sub>''i''</sub> to those functions which are zero for all arguments but ''i''. Now let ''M'' be an arbitrary ''R''-module and ''f''<sub>''i''</sub> : ''M''<sub>''i''</sub> → ''M'' be arbitrary ''R''-linear maps for every ''i'', then there exists precisely one ''R''-linear map

:<math>f : \bigoplus_{i \in I} M_i \rightarrow M</math>

such that ''f'' o ''j<sub>i</sub>'' = ''f''<sub>''i''</sub> for all ''i''.