|
Let
 be a collection of modules in some category of modules. Then the direct sum
 of that collection is the submodule of the direct product of the  consisting of all elements  such that all but a finite number of the  are zero.
For each  we have a projection
 defined by
 , and an injection
 where an element  of  maps to the element of
 whose  th term is  and every other term is zero.
The direct sum
 satisfies a certain universal property. Namely, if  is a module and there exist homomorphisms
 for all  , then there exists a unique homomorphism
 satisfying
 for all  .
The direct sum is often referred to as the weak direct sum or simply the sum.
Compare this to the direct product of modules.
Often an internal direct sum is written as
 .
| 
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ X_i & & Y \ar[ll]_{f_i} \ & \coprod_{i \in I} X_i \ar[ul]^{p_i} \ar@{-->}[ur]_{\phi} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/1360/l2h/img22.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ X_i & & Y \ar[ll]_{f_i} \ & \coprod_{i \in I} X_i \ar[ul]^{p_i} \ar@{-->}[ur]_{\phi} } } \end{xy}$")