# Definition:Free Module on Set/Canonical Basis

## Definition

Let $R$ be a ring with unity.

Let $\displaystyle R^{\left({I}\right)} = \bigoplus_{i \mathop \in I} R$ be the free $R$-module on $I$.

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 $\left\{ {e_j}\right\}_{j \mathop \in I}$.