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 $I$.

Let $M = \displaystyle \prod_{i \mathop \in I} M_i$ be their direct product.

The direct sum $\displaystyle \bigoplus_{i \mathop \in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support.

In Direct Sum of Modules is Module, it is shown that the direct sum is indeed a module.

Examples

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

Also see

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