Canonical Basis of Free Module on Set is Basis

Theorem
Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{\paren I}$ be the free $R$-module on $I$.

Let $B$ be its canonical basis.

Then $B$ is a basis of $R^{\paren I}$.

Proof
Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{\paren I}$ be the free $R$-module on $I$.

Let $B$ be its canonical basis.

Recall the definition of canonical basis:

It is to be shown that $B$ is a basis.

Hence the result.