direct sum

Let $\{X_i:i\in I\}$ be a collection of modulesin some category of modules.Then the direct sum ${\coprod }_{i\in I}{X}_i$of that collection is the submoduleof the direct product (http://planetmath.org/DirectProduct) of the $X_i$consisting of all elements $(x_i)$such that all but a finite numberof the $x_i$ are zero.

For each $j\in I$ we havea projection $p_j:{\coprod }_{i\in I}{X}_i\to X_j$defined by $(x_i)\mapsto x_j$,andan injection ${\lambda }_j:X_j\to {\coprod }_{i\in I}{X}_i$where an element $x_j$ of $X_j$maps to the element of ${\coprod }_{i\in I}{X}_i$whose $j$th term is $x_j$ and every other term is zero.

The direct sum ${\coprod }_{i\in I}{X}_i$satisfies a certain universal property.Namely, if $Y$ is a moduleand there exist homomorphisms $f_i:Y\to X_i$for all $i\in I$,then there exists a unique homomorphism$\varphi :{\coprod }_{i\in I}{X}_i\to Y$satisfying $p_i\varphi =f_i$ for all $i\in I$.

The direct sum is often referred toas the weak direct sumor simply the sum.

Compare this to the direct product of modules.

Often an internal direct sum is written as ${\oplus }_{i\in I}{X}_i$.