Universal Property of Free Module on Set
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Theorem
Let $R$ be a ring with unity.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Let $M$ be an $R$-module.
Let $\family {m_i}_{i \mathop \in I}$ be a family of elements of $M$.
Then there exists a unique $R$-module morphism:
- $\Psi: R^{\paren I} \to M$
that sends the $i$th canonical basis element to $m_i$, for all $i \in I$.
Moreover:
- $\ds \map \Psi {\family {r_i}_{i \mathop \in I} } = \sum_{i \mathop \in I} r_i m_i$
Proof
Existence
By Morphism from Ring with Unity to Module, for all $i$ there exists a morphism:
- $\psi_i: R \to M$
with $\map {\psi_i} 1 = m_i$.
By Universal Property of Direct Sum of Modules, there exists a morphism:
- $\Psi: R^{\paren I} \to M$
such that $\Psi \circ \iota_i = \psi_i$ for all $i$.
Thus:
- $\forall i \in I: \map \Psi {e_i} = \map \Psi {\map {\iota_i} 1} = \map {\psi_i} 1 = m_i$
We have:
- $\ds \family {r_i}_{i \mathop \in I} = \sum_{i \mathop \in I} r_i e_i$
so the expression for $\Psi$ follows by linearity.
$\Box$
Uniqueness
Let $\Psi$ be such a morphism.
Then $\Psi \circ \iota_i$ sends $1$ to $m_i$.
By Morphism from Ring with Unity to Module:
- $\Psi \circ \iota_i = \psi_i$
with $\psi_i$ as above.
By Universal Property of Direct Sum of Modules, $\Psi$ is determined by $\Psi \circ \iota_i$.
Thus $\Psi$ is unique.
$\blacksquare$