Direct Sum of Modules is Module

Theorem
Let $A$ be a commutative ring with unity.

Let $I$ be an indexing set.

Let $\family {M_i}_{i \mathop \in I}$ be a family of $A$-modules indexed by $I$.

Let $\ds M = \bigoplus_{i \mathop \in I} M_i$ be their direct sum.

Then $M$ is an $A$-module.

Also see

 * Elements of Finite Support form Submagma of Direct Product