[[Module theory MOC]]
# Direct sum of modules

The **direct sum** $\bigoplus_{i \in I} V^i$ of (right) [[Module|modules]] $(V^i)_{i \in I}$ over $R$ is their [[Products and coproducts|coproduct]] in $\lMod R$. #m/def/linalg 
It may be constructed as tuples with componentwise addition.

## Internal direct sum

$V$ is the **direct sum** $\bigoplus_{i\in I} S_{i}$ iff every $v \in V$ may be written as the sum of finitely many $v_{i} \in S_{i}$ in precisely one way.

## See also

- [[Graded module]]

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