Direct sum of modules

The direct sum $⨁ 𝑖 \in 𝐼 𝑉 𝑖$ of (right) modules $(𝑉 𝑖) 𝑖 \in 𝐼$ over $𝑅$ is their coproduct in $𝑅 𝖬 𝗈 𝖽$ . #m/def/linalg It may be constructed as tuples with componentwise addition.

Internal direct sum

$𝑉$ is the direct sum $⨁ 𝑖 \in 𝐼 𝑆 𝑖$ iff every $𝑣 \in 𝑉$ may be written as the sum of finitely many $𝑣 𝑖 \in 𝑆 𝑖$ in precisely one way.

Direct sum of modules

Internal direct sum

See also