Definition:Internal Direct Sum of Modules/Definition 3

Definition

Let $R$ be a ring.

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

Let $\left\langle{M_i}\right\rangle_{i \mathop \in I}$ be a family of submodules.

Let $\displaystyle \bigoplus_{i \mathop \in I} M_i$ be the external direct sum of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$.

$M$ is the internal direct sum of $\left\langle{M_i}\right\rangle_{i \mathop \in I}$ if and only if the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$.