\begin{definition}[Definition:Internal Direct Sum of Modules/Definition 3]
Let $R$ be a ring.
Let $M$ be an $R$-module.
Let $\family {M_i}_{i \mathop \in I}$ be a family of submodules.
Let $\ds \bigoplus_{i \mathop \in I} M_i$ be the external direct sum of $\family {M_i}_{i \mathop \in I}$.
$M$ is the '''internal direct sum''' of $\family {M_i}_{i \mathop \in I}$ {{iff}} the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$.
Category:Definitions/Internal Direct Sums
Category:Definitions/Linear Algebra
\end{definition}