\begin{definition}[Definition:Free Module on Set/Canonical Basis]
Let $R$ be a ring with unity.
Let $\ds R^{\paren I} = \bigoplus_{i \mathop \in I} R$ be the free $R$-module on $I$.
The '''$j$th canonical basis element''' is the element
:$e_j = \sequence {\delta_{ij} }_{i \mathop \in I} \in R^{\paren I}$
where $\delta$ denotes the Kronecker delta.
The '''canonical basis''' of $R^{\paren I}$ is the indexed set $\family {e_j}_{j \mathop \in I}$.
\end{definition}