Definition:Internal Direct Sum of Modules/Definition 2

Definition
Let $R$ be a ring.

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

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

$M$ is the internal direct sum of $\sequence {M_i}_{i \mathop \in I}$ :
 * $\ds \bigcup_{i \mathop \in I} M_i$ generates $M$
 * For all $i \in I$, $M_i \cap \ds \sum_{j \mathop \ne i} M_j = \set 0$