# Definition:Internal Direct Sum of Modules

## Definition

Let $R$ be a ring.

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

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

### Definition 1

$M$ is the internal direct sum of $(M_i)_{i\in I}$ if and only if every $m\in M$ can be written uniquely as a sum $\sum m_i$ with each $m_i\in M_i$.

### Definition 2

$M$ is the internal direct sum of $(M_i)_{i\in I}$ if and only if:

• $\displaystyle\bigcup_{i\in I}M_i$ generates $M$
• For all $i\in I$, $M_i\cap \displaystyle\sum_{j\neq i} M_j = \{0\}$

### Definition 3

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 $M$.