992: Characteristic of Ring

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definition of characteristic of 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: Note

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Starting Context

• The reader knows a definition of ring.

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Target Context

• The reader will have a definition of characteristic of 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:
$R$: $\in \{\text{ the rings }\}$
$\ast Ch(R)$: $\in \mathbb{N}$
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Conditions:
$Ch(R)\cdot 1=0\wedge \mathrm{\forall }n\in \mathbb{N}\setminus \{0\}\text{ such that }n<Ch(R)(n\cdot 1\ne 0)$
$\vee$
$Ch(R)=0$
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2: Note

In other words, $Ch(R)$ is the smallest positive natural number such that $Ch(R)$ times $1$ is $0$ if there is such, or $0$ if there is no such.

$Ch(R)=1$ means that $1\cdot 1=1=0$, which implies that $R=\{0=1\}$, because for each $r\in R$, $r=r1=r0=0$.

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References

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