Definition:Canonical Basis of Free Module on Set

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 indexed by $I$.

The $j$th canonical basis element $e_j$ is the element $\left \langle{r_i}\right\rangle_{i \mathop \in I} \in R^{\left({I}\right)}$ with $r_i = 1$ if $i = j$ and $0$ otherwise.

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