Free Module on Set is Free
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Theorem
Let $R$ be a ring with unity.
Let $I$ be a set.
Let $R^{\paren I}$ be the free $R$-module on $I$.
Then $R^{\paren I}$ is a free $R$-module.
Proof
From Canonical Basis of Free Module on Set is Basis, $R^{\paren I}$ has a basis.
$\blacksquare$