16D10-DirectProductOfModules.tex

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\pmtitle{direct product of modules}
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\pmtype{Definition}
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\pmsynonym{strong direct sum}{DirectProductOfModules}
\pmsynonym{complete direct sum}{DirectProductOfModules}
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%%%\usepackage{xypic}
\begin{document}
Let $\{ X_i : i \in I \}$ be a collection of modules
in some category of modules.
Then the {\it 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 {\it projection} $p_j : \prod_{i \in I} X_i \to X_j$
defined by $(x_i) \mapsto x_j$,
and
an {\it 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 {\it complete direct sum},
or the {\it strong direct sum},
or simply the {\PMlinkescapetext {\it product}}.

Compare this to the direct sum of modules.
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\end{document}