Direct product of modules

The product of modules: all tuples with coordinatewise operations.

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: giving a homomorphism $X\to \prod_i M_i$ is equivalent to giving compatible homomorphisms $X\to M_i$ for all $i$ .

For infinite $I$ , the product is strictly larger than the direct sum because it allows infinitely many nonzero coordinates.

Examples:

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