Definition:Internal Direct Sum of Modules/Definition 1
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Definition
Let $R$ be a ring.
Let $M$ be an $R$-module.
Let $\family {M_i}_{i \mathop \in I}$ be a family of submodules.
$M$ is the internal direct sum of $\family {M_i}_{i \mathop \in I}$ if and only if every $m \in M$ can be written uniquely as a summation $\ds \sum m_i$ with each $m_i \in M_i$.