direct product of modules

Let $\{X_i:i\in I\}$ be a collection of modulesin some category of modules.Then the direct product ${\prod }_{i\in I}{X}_i$of that collection is the modulewhose underlying set is the Cartesian product of the $X_i$with componentwise addition and scalar multiplication.For example, in a category of left modules:

$$(x_i)+(y_i)=(x_i+y_i),$$

$$r(x_i)=(r{x}_i).$$

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

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

The direct product is often referred toas the complete direct sum,or the strong direct sum,or simply the .

Compare this to the direct sum of modules.