631: Principal Integral Domain

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definition of principal integral domain

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Topics

About: ring

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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 integral domain.
• The reader knows a definition of principal ideal of ring.

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Target Context

• The reader will have a definition of principal integral domain.

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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:
$\ast R$: $\in \{\text{ the integral domains }\}$
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Conditions:
$\mathrm{\forall }I\in \{\text{ the ideals of }R\}(I\in \{\text{ the principal ideals of }R\})$
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2: Natural Language Description

Any integral domain, $R$, such that its each ideal is a principal ideal

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3: Note

It is called also "Principal Ideal Domain". In fact, "Principal Ideal Integral Domain" will be more appropriate.

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References

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