643: For Ring, Multiple of 0 Is 0

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description/proof of that for ring, multiple of 0 is 0

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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 ring.

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Target Context

• The reader will have a description and a proof of the proposition that for any ring, any multiple of 0 is 0.

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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:
$R$: $\in \{\text{ the rings }\}$
$p$: $\in R$
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Statements:
$0p=p0=0$.
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2: Natural Language Description

For any ring, $R$, and any element, $p\in R$, $0p=p0=0$.

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3: Proof

$0p+p=0p+1p=(0+1)p=1p=p$, which implies that $0p=0p+p-p=p-p=0$.

$p0+p=p0+p1=p(0+1)=p1=p$, which implies that $p0=p0+p-p=p-p=0$.

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References

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