977: For Ring, if Element Has Inverse, Inverse Is Unique

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description/proof of that for ring, if element has inverse, inverse is unique

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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: 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, if an element has an inverse, the inverse is unique.

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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:
$R$: $\in \{\text{ the rings }\}$
$r$: $\in R$
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Statements:
$\mathrm{\exists }{r}^{\prime }\in R(r^{\prime }r=r{r}^{\prime }=1)\wedge \mathrm{\exists }{r}^{″}\in R(r^{″}r=r{r}^{″}=1)$
$⟹$
$r^{\prime }=r^{″}$
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2: Proof

Whole Strategy: Step 1: multiply $r{r}^{″}=1$ by $r^{\prime }$ from left, and see that $r^{″}=r^{\prime }$.

Step 1:

From $r{r}^{″}=1$, $r^{\prime }r{r}^{″}=r^{\prime }1=r^{\prime }$.

The left hand side is $r^{\prime }r{r}^{″}=(r^{\prime }r)r^{″}=1r^{″}=r^{″}$.

So, $r^{″}=r^{\prime }$.

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References

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