Definition:Direct Sum of Modules

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

Let $\{M_i\}_{i \in I}$ be a collection of $A$-modules.

The direct sum $\displaystyle\bigoplus_{i \in I} M_i$ is the submodule of the direct product $\displaystyle\prod_{i \in I} M_i$ consisting of families $(m_i)_{i \in I}$ such that only finitely many of the $m_i$ are non-zero.