direct product of modules

Let $\{X_{i}:i\in I\}$ be a collection of modules in some category of modules. Then the direct product $\prod_{i\in I}X_{i}$ 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 projection $p_{j}:\prod_{i\in I}X_{i}\to X_{j}$ defined by $(x_{i})\mapsto x_{j}$ , and an 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 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
Canonical name	DirectProductOfModules
Date of creation	2013-03-22 12:09:34
Last modified on	2013-03-22 12:09:34
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	10
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 16D10
Synonym	strong direct sum
Synonym	complete direct sum
Related topic	CategoricalDirectProduct
Defines	direct product