626: Basis of Module

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definition of basis of module

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Topics

About: module

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of %ring name% module.
• The reader knows a definition of linearly independent subset of module.

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Target Context

• The reader will have a definition of basis of module.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the rings }\}$
$M$: $\in \{\text{ the modules over }R\}$
$\ast B$: $\subseteq M$, $\in \{\text{ the (possibly uncountable) linearly independent subsets of }M\}$
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Conditions:
$\mathrm{\forall }p\in M(\mathrm{\exists }S\in \{\text{ the finite subsets of }B\},\mathrm{\exists }{r}^j\in R(p=\sum _{b_j\in S}{r}^j{b}_j))$
//

$S$ 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.

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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 $B$

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3: Note

As any vectors space is a module, 'basis of vectors space' is nothing but 'basis of module'.

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References

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