# Universal Property of Direct Sum of Modules

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## Theorem

Let $R$ be a ring.

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

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

Let $M = \displaystyle \bigoplus_{i \mathop \in I} M_i$ be their direct sum.

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

Then there exists a unique morphism:

- $\Psi: M \to N$

such that:

- $\forall i: \psi_i = \Psi \circ \iota_i$

where $\iota_i: M_i \to M$ is the $i$th canonical injection.

## Proof