Universal Property of Direct Product of Modules

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

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.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Universal Property of Direct Sum of Modules