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 $(r_i)_{i\in I}\in\displaystyle R^{\left({I}\right)}$ with $r_i=1$ if $i=j$ and $0$ otherwise.

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