Definition:Internal Direct Sum of Modules/Definition 2

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 $(M_i)_{i\in I}$ :
 * $\displaystyle\bigcup_{i\in I}M_i$ generates $M$
 * For all $i\in I$, $M_i\cap \displaystyle\sum_{j\neq i} M_j = \{0\}$