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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 self-studying abstract algebra from the book "Abstract Algebra An Inquiry-Based 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? Thanks from edv April 27th, 2015, 09:15 AM   #5
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 In the integers mod 6, 1 is a unit. Is 1 + 1 a unit?
No, it isn't: + =  and there are no such integers x and n that x * 2 = 6 * n + 1

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. Thanks from edv April 27th, 2015, 09:27 AM   #7
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 Originally Posted by Country Boy 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.
Of course! I didn't see this. Thanks! 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. Tags problem, ring, units Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ricsi046 Abstract Algebra 6 March 17th, 2014 10:14 AM danimeck Abstract Algebra 2 February 22nd, 2014 11:15 AM brisunimath Abstract Algebra 2 February 23rd, 2013 07:42 AM HairOnABiscuit Abstract Algebra 1 November 23rd, 2009 11:13 AM sillyme Abstract Algebra 16 February 10th, 2009 05:36 PM

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