description/proof of that for ring, if element has inverse, inverse is unique
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
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.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(R\): \(\in \{\text{ the rings }\}\)
\(r\): \(\in R\)
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Statements:
\(\exists r' \in R (r' r = r r' = 1) \land \exists r'' \in R (r'' r = r r'' = 1)\)
\(\implies\)
\(r' = r''\)
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2: Proof
Whole Strategy: Step 1: multiply \(r r'' = 1\) by \(r'\) from left, and see that \(r'' = r'\).
Step 1:
From \(r r'' = 1\), \(r' r r'' = r' 1 = r'\).
The left hand side is \(r' r r'' = (r' r) r'' = 1 r'' = r''\).
So, \(r'' = r'\).
