Definition:Internal Direct Sum of Modules/Definition 1

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 $\family {M_i}_{i \mathop \in I}$ every $m \in M$ can be written uniquely as a summation $\ds \sum m_i$ with each $m_i \in M_i$.