647: Irreducible Element of Commutative Ring

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definition of irreducible element of commutative ring

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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

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Starting Context

• The reader knows a definition of ring.
• The reader knows a definition of units of ring.

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Target Context

• The reader will have a definition of irreducible element of commutative ring.

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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:
$R$: $\in \{\text{ the commutative rings }\}$
$U$: $=\{\text{ the units of }R\}$
$\ast p$: $\in R$
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Conditions:
$p\ne 0$
$\wedge$
$p\notin U$
$\wedge$
$\mathrm{\exists }{p}_1,p_2\in R\text{ such that }p=p_1{p}_2⟹(p_1\in U\vee p_2\in U)$
//

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2: Natural Language Description

For any commutative ring, $R$, and the set of the units of $R$, $U$, any element, $p\in R$, such that if there are any $p_1,p_2\in R$ such that $p=p_1{p}_2$, $p_1\in U$ or $p_2\in U$

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References

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