Definition:Free Module on Set/Canonical Basis

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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