\begin{definition}[Definition:Internal Direct Sum of Modules/Definition 2]
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 $\sequence {M_i}_{i \mathop \in I}$ {{iff}}:
:$\ds \bigcup_{i \mathop \in I} M_i$ generates $M$
:For all $i \in I$, $M_i \cap \ds \sum_{j \mathop \ne i} M_j = \set 0$
Category:Definitions/Internal Direct Sums
Category:Definitions/Linear Algebra
\end{definition}