# Definition:Free Module on Set

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## 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 $e_i$.

## Also see