Canonical Basis of Free Module on Set is Basis

Theorem
Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{(I)}$ be the free $R$-module on $I$.

Let $B$ be its canonical basis.

Then $B$ is a basis of $R^{(I)}$.

Proof
Suppose $R$ is a ring with unity.

Suppose $I$ is a set.

Suppose $R^{(I)}$ is the free $R$-module on $I$.

Suppose $B$ is the canonical basis for $R^{(I)}$.

Then by the definition of canonical basis of $R^{(I)}$, the $j$th canonical basis element is the element:
 * $e_j = \family {\delta_{i j} }_{i \mathop \in I} \in R^{\paren I}$

where $\delta$ denotes the Kronecker delta.

Moreover, by the definition of canonical basis, it follows that the canonical basis of $R^{\paren I}$ is the indexed family $\family {e_j}_{j \mathop \in I}$.

It remains to be shown that the canonical basis of $R^{\paren I}$ is a basis.

Hence, it follows that the canonical basis of $R^{\paren I}$ is a basis.