Definition:Free Module on Set
(Redirected from Definition:Free Vector Space on Set)
Jump to navigation
Jump to search
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