# Definition:Free Module on Set

## Definition

Let $R$ be a ring.

Let $I$ be an indexing set.

The free $R$-module on $I$ is the direct sum of $R$ as a module over itself:

$\displaystyle R^{\left({I}\right)} := \bigoplus_{i \mathop \in I} R$

of the family $I \to \{R\}$ to the singleton $\{R\}$.

### Canonical Basis

Let $R$ be a ring with unity.

The $j$th canonical basis element is the element

$e_j=\left \langle{\delta_{ij}}\right\rangle_{i \mathop \in I} \in R^{\left({I}\right)}$

where $\delta$ denotes the Kronecker delta.

The canonical basis of $R^{\left({I}\right)}$ is the indexed set $\left\{ {e_j}\right\}_{j \mathop \in I}$.

### Canonical Mapping

The canonical mapping $I \to R^{(I)}$ is the mapping that sends $i \in I$ to the $i$th standard basis element $e_i$.