Direct sum of modules

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

\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: for every $R$ -module $N$ , an arbitrary family of homomorphisms $f_i:M_i\to N$ determines a unique homomorphism $\bigoplus_iM_i\to N$ .

Examples

For a finite index set, $\bigoplus_{i=1}^n M_i=\prod_{i=1}^n M_i$ .
$\bigoplus_{n\ge 1}\mathbb Z$ consists of integer sequences with finite support.
If $I=\varnothing$ , then $\bigoplus_{i\in I}M_i$ is the zero module.