Direct sum of modules

The coproduct of modules: tuples with finite support under coordinatewise operations.

Direct sum of modules

Given a family of $R$ -modules $(M_i)_{i\in I}$ , their direct sum is the set

\bigoplus_{i\in I} M_i=\{(m_i)\in \prod_{i\in I} M_i : m_i=0 \text{ for all but finitely many } i\},

with coordinatewise addition and scalar multiplication. It is naturally a submodule of the direct product , which itself is modeled on the Cartesian product of sets.

The direct sum is characterized by the universal property of the direct sum : maps out of a direct sum are uniquely determined by maps out of each summand, subject to finite support.

Examples:

For a finite index set, $\bigoplus_{i=1}^n M_i$ is the same as $\prod_{i=1}^n M_i$ .
$\bigoplus_{n\ge 1}\mathbb Z$ consists of integer sequences with only finitely many nonzero entries.
(Edge case) If $I=\varnothing$ , then $\bigoplus_{i\in I} M_i$ is the zero module.