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

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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:

The canonical basis of $R^{\paren I}$ is the indexed family $\family {e_j}_{j \mathop \in I}$, defined as:

$e_j = \family {\delta_{i j} }_{i \mathop \in I} \in R^{\paren I}$

where:

$\delta$ denotes the Kronecker delta.

$e_j$ is known as the $j$th canonical basis element of $R^{\paren I}$.

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

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Hence the result.