644: Cancellation Rule on Integral Domain

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description/proof of cancellation rule on 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: Proof

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Starting Context

• The reader knows a definition of integral domain.

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Target Context

• The reader will have a description and a proof of the proposition that the cancellation rule holds on any 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:
$R$: $\in \{\text{ the integral domains }\}$
$p_1$: $\in R$
$p_2$: $\in R$
$p_3$: $\in R$
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Statements:
(
$p_1\ne 0$
$\wedge$
$p_1{p}_2=p_1{p}_3$
)
$⟹$
$p_2=p_3$
//

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2: Natural Language Description

For any integral domain, $R$, and any elements, $p_1,p_2,p_3\in R$, if $p_1\ne 0$ and $p_1{p}_2=p_1{p}_3$, $p_2=p_3$.

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3: Proof

Let us suppose that $p_1\ne 0$ and $p_1{p}_2=p_1{p}_3$.

$p_1(p_2-p_3)=0$. As $p_1\ne 0$, $p_2-p_3=0$, by the definition of integral domain. So, $p_2=p_3$.

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References

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