Direct product of modules

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

\prod_{i\in I} M_i=\{(m_i)_{i\in I}: m_i\in M_i\},

with coordinatewise addition and scalar multiplication. As a set it is the Cartesian product, and it satisfies the categorical product universal property: for every $R$ -module $X$ , giving a homomorphism $X\to \prod_i M_i$ is equivalent to giving a homomorphism $X\to M_i$ for each $i$ .

Remarks

For infinite $I$ , the product contains the direct sum and may also contain tuples with infinitely many nonzero coordinates. For example, $\bigoplus_{n\ge1}\mathbb Z$ is strictly contained in $\prod_{n\ge1}\mathbb Z$ .

Examples

$\prod_{n\ge 1}\mathbb Z$ is the set of all integer sequences, with no finiteness restriction.
For modules $M,N$ , the product $M\times N$ is the usual binary product with projections.
If each $M_i=0$ , then the product is $0$ , even if $I$ is infinite.