# Definition:Free Module on Set

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## Definition

Let $R$ be a ring.

This article, or a section of it, needs explaining.In particular: For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, that is, complete with operators, $\struct {R, +, \circ}$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

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