628: Finitely-Generated Module

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definition of finitely-generated 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 generator of module.

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Target Context

• The reader will have a definition of finitely-generated 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:
$R$: $\in \{\text{ the rings }\}$
$\ast M$: $\in \{\text{ the modules over }R\}$
//

Conditions:
$\mathrm{\exists }S\in \{\text{ the finite generators of }M\}$
//

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2: Natural Language Description

For any ring, $R$, any module, $M$, that has a finite generator

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3: Note

Being finitely-generated does not imply having any basis: see Note of the definition of generator of module.

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References

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