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

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