Definition:Canonical Basis of Free Module on Set

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 canonical basis of $R^{\paren I}$ is the indexed family $\family {e_j}_{j \mathop \in I}$, defined as:
 * $e_j = \family {\delta_{i j} }_{i \mathop \in I} \in R^{\paren I}$

where:
 * $\delta$ denotes the Kronecker delta.
 * $e_j$ is known as the $j$th canonical basis element of $R^{\paren I}$.

Also see

 * Canonical Basis of Free Module on Set is Basis