\begin{definition}[Definition:Internal Direct Sum of Modules/Definition 1]
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 $\family {M_i}_{i \mathop \in I}$ {{iff}} every $m \in M$ can be written uniquely as a summation $\ds \sum m_i$ with each $m_i \in M_i$.
Category:Definitions/Internal Direct Sums
Category:Definitions/Linear Algebra
\end{definition}