Definition:Free Module on Set
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Definition
Let $R$ be a ring.
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Let $I$ be an indexing set.
The free $R$-module on $I$ is the direct sum of $R$ as a module over itself:
- $\ds R^{\paren I} := \bigoplus_{i \mathop \in I} R$
of the family $I \to \set R$ to the singleton $\set R$.
Canonical Basis
Let $R$ be a ring with unity.
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 $\family {e_j}_{j \mathop \in I}$.
Canonical Mapping
The canonical mapping $I \to R^{\paren I}$ is the mapping that sends $i \in I$ to the $i$th standard basis element $e_i$.
Also see