# direct product of modules

Let $\{X_{i}:i\in I\}$ be a collection of modules in some category of modules. Then the $\prod_{i\in I}X_{i}$ of that collection is the module whose underlying set is the Cartesian product of the $X_{i}$ with componentwise addition and scalar multiplication. For example, in a category of left modules:

 $(x_{i})+(y_{i})=(x_{i}+y_{i}),$
 $r(x_{i})=(rx_{i}).$

For each $j\in I$ we have a $p_{j}:\prod_{i\in I}X_{i}\to X_{j}$ defined by $(x_{i})\mapsto x_{j}$, and an $\lambda_{j}:X_{j}\to\prod_{i\in I}X_{i}$ 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 module and there exist homomorphisms $f_{i}:X_{i}\to Y$ for all $i\in I$, then there exists a unique homomorphism $\phi:Y\to\prod_{i\in I}X_{i}$ satisfying $\phi\lambda_{i}=f_{i}$ for all $i\in I$.

 $\xymatrix{X_{i}\ar[dr]_{\lambda_{i}}\ar[rr]^{f_{i}}&&Y\ar@{-->}[dl]^{\phi}\\ &\prod_{i\in I}X_{i}}$

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

Compare this to the direct sum of modules.

Title direct product of modules DirectProductOfModules 2013-03-22 12:09:34 2013-03-22 12:09:34 Mathprof (13753) Mathprof (13753) 10 Mathprof (13753) Definition msc 16D10 strong direct sum complete direct sum CategoricalDirectProduct direct product