Definition:Direct Sum of Modules
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Definition
Let $A$ be a commutative ring with unity.
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Let $\family {M_i}_{i \mathop \in I}$ be a family of $A$-modules indexed by $I$.
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Let $M = \ds \prod_{i \mathop \in I} M_i$ be their direct product.
The direct sum $\ds \bigoplus_{i \mathop \in I} M_i$ is the submodule of $M$ the consisting of the elements of finite support.
Examples
- A particular case is that of a free module on a set.