Direct Sums and Products

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

\[ \prod_{i\in I} M_i = \{ (x_i)_{i\in I} | x_i \in M_i \} \]

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 $M_j \cap (\sum_{i \ne j} M_i) = \{ 0 \}$.