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Homedirect product of modules

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# 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 product.

Compare this to the direct sum of modules.

## Mathematics Subject Classification

16D10*no label found*

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