definition of basis of module
Topics
About: module
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %ring name% module.
- The reader knows a definition of linearly independent subset of module.
Target Context
- The reader will have a definition of basis of module.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( M\): \(\in \{\text{ the modules over } R\}\)
\(*B\): \(\subseteq M\), \(\in \{\text{ the (possibly uncountable) linearly independent subsets of } M\}\)
//
Conditions:
\(\forall p \in M (\exists S \in \{\text{ the finite subsets of } B\}, \exists r^j \in R (p = \sum_{b_j \in S} r^j b_j))\)
//
\(S\) has to be a finite subset of \(B\), because otherwise, \(p = \sum_{b_j \in S} r^j b_j\) would not make sense without \(M\) equipped with any norm: definition of convergence of infinite series requires a norm.
2: Natural Language Description
For any ring, \(R\), and any module, \(M\), over \(R\), any (possibly uncountable) linearly independent subset, \(B \subseteq M\), such that each element of \(M\) is a linear combination of some (finite) elements of \(B\)
3: Note
As any vectors space is a module, 'basis of vectors space' is nothing but 'basis of module'.
