Commutative Algebra - 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 $M_{j} \cap (\sum_{i \neq j} M_{i}) = {0}$ .