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direct product of modules (Definition)

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:

$\displaystyle (x_i) + (y_i) = (x_i + y_i),$
$\displaystyle r (x_i) = (r x_i).$

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 $ \phi : Y \to \prod_{i \in I} X_i$ satisfying $ \phi \lambda_i = f_i$ for all $ i \in I$.

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ X_i \ar[dr]_{\lambda_i} \ar[rr]^{f_i} & & Y \ar@{-->}[dl]^{\phi} \ & \prod_{i \in I} X_i } } \end{xy}$

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.



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See Also: categorical direct product

Other names:  strong direct sum, complete direct sum
Also defines:  direct product
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Cross-references: direct sum, homomorphisms, universal property, term, maps, injection, projection, left modules, multiplication, scalar, addition, Cartesian product, category, modules, collection
There are 10 references to this entry.

This is version 6 of direct product of modules, born on 2002-01-05, modified 2007-01-06.
Object id is 1357, canonical name is DirectProduct.
Accessed 5183 times total.

Classification:
AMS MSC16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)

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