Definition:Internal Direct Sum of Modules

Definition

Let $R$ be a ring.

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

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

Definition 1

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

Definition 2

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

$\ds \bigcup_{i \mathop \in I} M_i$ generates $M$

For all $i \in I$, $M_i \cap \ds \sum_{j \mathop \ne i} M_j = \set 0$

Definition 3

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 $\family {M_i}_{i \mathop \in I}$ if and only if the mapping given by Universal Property of Direct Sum of Modules is an isomorphism onto $M$.

Also see

• Equivalence of Definitions of Internal Direct Sum of Modules
• Definition:Internal Direct Sum of Vector Spaces