648: Unique Factorization Domain

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definition of unique factorization 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 irreducible element of commutative ring.

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Target Context

• The reader will have a definition of unique factorization 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:
$\ast R$: $\in \{\text{ the integral domains }\}$
$U$: $=\{\text{ the units of }R\}$
$I$: $=\{\text{ the irreducible elements of }R\}$
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Conditions:
$\mathrm{\forall }p\in R(\mathrm{\exists }u\in U,\mathrm{\exists }{i}_j\in I(p=u{i}_1...i_n))$
$\wedge$
(
$\mathrm{\exists }{u}^{\prime }\in U,\mathrm{\exists }{i}_j^{\prime }\in U(p=u^{\prime }{i}_1^{\prime }...i_m^{\prime })$
$⟹$
$m=n\wedge \mathrm{\exists }f:\{i_1,...,i_n\}\to \{i_1^{\prime },...,i_n^{\prime }\}\in \{\text{ the bijections }\},\mathrm{\exists }{u}_j\in U(i_j^{\prime }=u_{j}f(i_j))$
)
//

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2: Natural Language Description

Any integral domain, $R$, such that for the set of the units of $R$, $U$, and the set of the irreducible elements of $R$, $I$, any element, $p\in R$, can be expressed as $p=u{i}_1...i_n$ for a $u\in U$ and an $i_j\in I$, and if $p=u^{\prime }{i}_1^{\prime }...i_m^{\prime }$ for a $u^{\prime }\in U$ and an $i_j^{\prime }\in I$, $m=n$ and there is a bijection, $f:\{i_1,...,i_n\}\to \{i_1^{\prime },...,i_n^{\prime }\}$, such that $i_j^{\prime }=u_{j}f(i_j)$ for a $u_j\in U$

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3: Note

This categorization is useful because some types of rings are known to be unique factorization domains: especially, any principal integral domain (of which each Euclidean domain (of which $\mathbb{Z}$ is) is) is a unique factorization domain.

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References

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