Definition:Direct Sum of Modules

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

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

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

Examples

 * A particular case is that of a free module indexed by a set.

Also see

 * Universal Property of Direct Sum of Modules
 * Definition:Module Direct Product