Definition:Internal Direct Sum of Modules/Definition 2

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Definition

Let $R$ be a ring.

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

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


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