Definition:Direct Sum of Modules

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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.