994: Characteristic of Integral Domain Is 0 or Prime Number

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description/proof of that characteristic of integral domain is 0 or prime number

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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: Note 1
• 3: Proof
• 4: Note 2

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Starting Context

• The reader knows a definition of characteristic of ring.

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Target Context

• The reader will have a description and a proof of the proposition that the characteristic of any integral domain is 0 or a prime number.

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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:
$R$: $\in \{\text{ the integral domains }\}$
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Statements:
$Ch(R)=0\vee Ch(R)\in \{\text{ the prime numbers }\}$
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2: Note 1

Any field is an integral domain, and so, the characteristic of any field is $0$ or a prime number.

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3: Proof

Whole Strategy: Step 1: suppose that $Ch(R)\ne 0$; Step 2: suppose that $Ch(R)=mn$ where $m,n\in \mathbb{N}\setminus \{0,1\}$, and find a contradiction.

Step 1:

Let us suppose that $Ch(R)\ne 0$.

Step 2:

Let us suppose that $Ch(R)$ was not any prime number.

That would mean that $Ch(R)=mn$ where $m,n\in \mathbb{N}\setminus \{0,1\}$.

$(mn)\cdot 1=(m\cdot 1)(n\cdot 1)=0$, by the distribution law.

$m\cdot 1=0$ or $n\cdot 1=0$, a contradiction against the supposition that $mn$ such that $m,n<mn$ was the smallest such.

So, $Ch(R)$ is a prime number.

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4: Note 2

$(m\cdot 1)(n\cdot 1)$ and $m\cdot (n\cdot 1)$ mean some different things, but they have the same value after all because of the distribution law: $(m\cdot 1)(n\cdot 1)=(1+...+1)(1+...+1)$ while $m\cdot (n\cdot 1)=(1+...+1)+...+(1+...+1)$.

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References

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