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April 26th, 2015, 12:21 AM   #1
edv
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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
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April 26th, 2015, 12:08 PM   #2
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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".
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April 26th, 2015, 10:10 PM   #3
edv
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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.
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April 27th, 2015, 07:58 AM   #4
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In the integers mod 6, 1 is a unit. Is 1 + 1 a unit?
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April 27th, 2015, 09:15 AM   #5
edv
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Quote:
In the integers mod 6, 1 is a unit. Is 1 + 1 a unit?
No, it isn't: [1]+[1] = [2] and there are no such integers x and n that x * 2 = 6 * n + 1

Thanks for counterexample that disproves it!
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April 27th, 2015, 09:19 AM   #6
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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.
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April 27th, 2015, 09:27 AM   #7
edv
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Quote:
Originally Posted by Country Boy View Post
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!
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April 27th, 2015, 01:24 PM   #8
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Simple example: in the ring of integers 1 is a unit but 1+1=2 is not.
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