# Universal Property of Direct Product of Modules

## Theorem

Let $R$ be a ring.

Let $N$ be an $R$-module.

Let $\family{M_i}_{i \mathop \in I}$ be a family of $R$-modules.

Let $M = \ds \prod_{i \mathop \in I} M_i$ be their direct product.

Let $\family{\psi_i}_{i \mathop \in I}$ be a family of $R$-module morphisms $N \to M_i$.

Then there exists a unique morphism:

$\Psi: N \to M$

such that:

$\forall i: \psi_i = \pi_i \circ \Psi$

where $\pi_i: M \to M_i$ is the $i$th canonical projection.