PlanetMath: direct product of modules

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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 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 of maps to the element of $\prod_{i \in I} X_i$ whose th term is and every other term is zero.

The direct product $\prod_{i \in I} X_i$ satisfies a certain universal property. Namely, if 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.

"direct product of modules" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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