direct sumDirectSum2002-01-05 15:28:012005-04-24 17:21:16Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}Let $\{ X_i : i \in I \}$ be a collection of modules
in some category of modules.
Then the {\it direct sum} $\coprod_{i \in I} X_i$
of that collection is the submodule
of the \PMlinkname{direct product}{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 {\it projection} $p_j : \coprod_{i \in I} X_i \to X_j$
defined by $(x_i) \mapsto x_j$,
and
an {\it 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
$\phi : \coprod_{i \in I} X_i \to Y$
satisfying $p_i \phi = f_i$ for all $i \in I$.
$$
\xymatrix{
X_i
&
&
Y
\ar[ll]_{f_i}
\\
&
\coprod_{i \in I} X_i
\ar[ul]^{p_i}
\ar@{-->}[ur]_{\phi}
}
$$
The direct sum is often referred to
as the {\it weak direct sum}
or simply the {\it sum}.
Compare this to the direct product of modules.
Often an internal direct sum is written as $\bigoplus_{i \in I} X_i$.