Definition:Internal Direct Sum of Modules/Definition 1

Definition
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 $(M_i)_{i\in I}$ every $m\in M$ can be written uniquely as a sum $\sum m_i$ with each $m_i\in M_i$.