Definition:Free Module on Set

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Let $R$ be a ring.

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

Special case