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November 24th, 2018, 09:21 AM  #1 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3  The unique multiplicative inverse
How can prove that the multiplicative inverse is unqiue?

November 24th, 2018, 09:32 AM  #2 
Senior Member Joined: Aug 2012 Posts: 2,329 Thanks: 720  
November 24th, 2018, 09:35 AM  #3 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3  two...
Why can't be 2 or 3 multiplactive identities?

November 24th, 2018, 09:40 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,467 Thanks: 1342  
November 24th, 2018, 09:54 AM  #5 
Senior Member Joined: Aug 2012 Posts: 2,329 Thanks: 720  1 isn't the usual 1. It's the symbol for the multiplicative identity (in a group, say). So if 2 were the multiplicative identity in some group, then 2 would be denoted as 1. Subtle point. But in general if $ab = ac$ and $a$ is invertible, then $b = c$ for the same reason as above. I don't even need the symbol 1 for this proof. 
November 24th, 2018, 10:05 AM  #6 
Senior Member Joined: Nov 2011 Posts: 250 Thanks: 3 
What will be if there is more than 1 such as two of or three of the multiplicative identity? What will happen then? What there is only and only one multiplicative identity? 
November 24th, 2018, 10:19 AM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 624 Thanks: 396 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
@OP: If you are actually trying to prove that the multiplicative identity is unique, then you should note that the invertible elements of a ring form a group under multiplication. So the identity is unique provided that the identity in a group is unique. This is essentially a 1 line proof which you should do as an exercise. Specifically, suppose $(G,*)$ is a group and $1, 1'$ are both identity elements. Prove that $1 = 1'$.  
November 25th, 2018, 08:33 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2643 Math Focus: Mainly analysis and algebra 
I presume that the said proof is something like if $1$ is an identity element then $1 * 1' = 1'$ and if $1'$ is an identity element then $1 * 1' = 1$ so if both are identity elements we have $$1 = 1 * 1' = 1'$$

November 25th, 2018, 04:14 PM  #9 
Senior Member Joined: Sep 2016 From: USA Posts: 624 Thanks: 396 Math Focus: Dynamical systems, analytic function theory, numerics  

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inverse, multiplicative, unique 
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