## Direct Sums and Products

Let $\{M_i | i\in I\}$ be a family of $R$-modules.
Define the **direct product** by

The direct product inherits an $R$-module structure by defining the operations coordinatewise.

**Direct sums** of modules
$\oplus_{i\in I} M_i$
are defined in the same way except only finitely many
of the coordinates of the tuples are nonzero.
Thus if $I$ is finite direct sums and products
are the same.

We say $M$ is the **internal direct sum** of the family if
$M = \sum M_i$ and
$M_j \cap (\sum_{i \ne j} M_i) = \{ 0 \}$.