Definition:Free Module on Set/Canonical Basis

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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

$e_j = \family {\delta_{i j} }_{i \mathop \in I} \in R^{\paren I}$

where $\delta$ denotes the Kronecker delta.

The canonical basis of $R^{\paren I}$ is the indexed family $\family {e_j}_{j \mathop \in I}$.

Also see