# direct sum

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

For each $j\in I$ we have a projection ${p}_{j}:{\coprod}_{i\in I}{X}_{i}\to {X}_{j}$ defined by $({x}_{i})\mapsto {x}_{j}$, and an 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 module
and 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 $i\in I$.

$$\text{xymatrix}{X}_{i}\mathrm{\&}\mathrm{\&}Y\text{ar}{[ll]}_{{f}_{i}}\mathrm{\&}\coprod _{i\in I}{X}_{i}\text{ar}{[ul]}^{{p}_{i}}\text{ar}\mathrm{@}-->{[ur]}_{\varphi}$$ |

The direct sum is often referred to as the weak direct sum or simply the sum.

Compare this to the direct product of modules.

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

Title | direct sum |
---|---|

Canonical name | DirectSum |

Date of creation | 2013-03-22 12:09:37 |

Last modified on | 2013-03-22 12:09:37 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 10 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16-00 |

Synonym | weak direct sum |

Related topic | CategoricalDirectSum |

Related topic | DirectSummand |

Related topic | DirectSumOfMatrices |