640: Associates of Element of Commutative Ring

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definition of associates of 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
• 3: Note

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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 associates of 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:
$R$: $\in \{\text{ the commutative rings }\}$
$U$: $=\{\text{ the units of }R\}$
$p$: $\in R$
$\ast Asc(p)$: $=\{up|u\in U\}$
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Conditions:
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2: Natural Language Description

For any commutative ring, $R$, the set of the units of $R$, $U$, and any element, $p\in R$, $Asc(p):=\{up|u\in U\}$

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3: Note

For each elements, $p,p^{\prime }\in R$, $p^{\prime }\in Asc(p)$, is an equivalence relation: 1) $p\in Asc(p)$, because $p=1p$; 2) $p_1\in Asc(p_2)⟹p_2\in Asc(p_1)$, because if $p_1=u{p}_2$, $p_2=u^{-1}{p}_1$; 3) $(p_1\in Asc(p_2)\wedge p_2\in Asc(p_3))⟹p_1\in Asc(p_3)$, because if $p_1=u_2{p}_2$ and $p_2=u_3{p}_3$, $p_1=u_2{u}_3{p}_3$ where $u_2{u}_3$ is a unit, because $u_3^{-1}{u}_2^{-1}{u}_2{u}_3=u_2{u}_3{u}_3^{-1}{u}_2^{-1}=1$.

So, $R/Asc$ is a quotient set.

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References

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