Definition:Free Module on Set/Canonical Basis

Definition
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 $\family {e_j}_{j \mathop \in I}$.

Also see

 * Canonical Basis of Free Module on Set is Basis