# 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:

$$({x}_{i})+({y}_{i})=({x}_{i}+{y}_{i}),$$ |

$$r({x}_{i})=(r{x}_{i}).$$ |

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
$\varphi :Y\to {\prod}_{i\in I}{X}_{i}$
satisfying $\varphi {\lambda}_{i}={f}_{i}$ for all $i\in I$.

$$\text{xymatrix}{X}_{i}\text{ar}{[dr]}_{{\lambda}_{i}}\text{ar}{[rr]}^{{f}_{i}}\mathrm{\&}\mathrm{\&}Y\text{ar}\mathrm{@}-->{[dl]}^{\varphi}\mathrm{\&}\prod _{i\in I}{X}_{i}$$ |

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 |