Definition:Internal Direct Sum of Modules/Definition 1

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Let $R$ be a ring.

Let $M$ be an $R$-module.

Let $(M_i)_{i\in I}$ be a family of submodules.

$M$ is the internal direct sum of $(M_i)_{i\in I}$ if and only if every $m\in M$ can be written uniquely as a sum $\sum m_i$ with each $m_i\in M_i$.