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April 26th, 2015, 12:21 AM  #1 
Newbie Joined: Apr 2015 From: Poland Posts: 4 Thanks: 0  Problem with units in ring
Hello! I'm new here so please don't laugh if I ask a trivial question. I'm selfstudying abstract algebra from the book "Abstract Algebra An InquiryBased Approach". Currently I'm stuck at one exercise about units in ring: Let $\displaystyle R$ be a ring with identity, and let $\displaystyle x$ and $\displaystyle y$ be units in $\displaystyle R$. Prove or disprove that $\displaystyle x+y$ is a unit in $\displaystyle R$. I've spent few hours and I have no idea how to prove (or disprove) that. Could you give me some clue? Best, edv 
April 26th, 2015, 12:08 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,804 Thanks: 715 
I am not familiar with this terminology. What is a "unit"? In a ring, the sum of any two members is a member (definition of a ring). If "unit" is the element x defined by x+y=y for any y (x is "zero"), then it holds for "unit". 
April 26th, 2015, 10:10 PM  #3 
Newbie Joined: Apr 2015 From: Poland Posts: 4 Thanks: 0 
As I remember the definition of a unit goes like this: $\displaystyle x$ is a unit in $\displaystyle R$ if and only if there exists such $\displaystyle y$ that $\displaystyle x * y = 1$ where 1 is the identity in $\displaystyle R$. Or in other words $\displaystyle x$ is a unit if there exists multiplicative inverse of $\displaystyle x$ in $\displaystyle R$ Last edited by edv; April 26th, 2015 at 10:12 PM. 
April 27th, 2015, 07:58 AM  #4 
Senior Member Joined: Aug 2012 Posts: 2,354 Thanks: 735 
In the integers mod 6, 1 is a unit. Is 1 + 1 a unit?

April 27th, 2015, 09:15 AM  #5  
Newbie Joined: Apr 2015 From: Poland Posts: 4 Thanks: 0  Quote:
Thanks for counterexample that disproves it!  
April 27th, 2015, 09:19 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Notice that if it were true that "if x and y are units then x+ y is a unit" then it would follow that any multiple of a unit would be a unit. Since every member of a ring is a multiple of 1, it would then follow that every member of a ring would be invertible and, of course, that is not true.

April 27th, 2015, 09:27 AM  #7  
Newbie Joined: Apr 2015 From: Poland Posts: 4 Thanks: 0  Quote:
 
April 27th, 2015, 01:24 PM  #8 
Global Moderator Joined: May 2007 Posts: 6,804 Thanks: 715 
Simple example: in the ring of integers 1 is a unit but 1+1=2 is not.


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