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Old 2016-04-08, 05:21   #1
wildrabbitt
 
Jul 2014

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Default Integral domain such that.....

Hi,


Let R be an integral domain in which every non-zero non-unit element has a factorisation into irreducible elements.

Can someone give me a concrete example of such a Ring?
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Old 2016-04-08, 08:00   #2
Nick
 
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Dec 2012
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Quote:
Hi,
Let R be an integral domain in which every non-zero non-unit element has a factorisation into irreducible elements.

Can someone give me a concrete example of such a Ring?
The set of all integers (under ordinary addition and multiplication) is an integral domain. Its units are 1 and -1, and its irreducible elements are:
2,-2,3,-3,5,-5,7,-7,11,-11,13,-13,17,-17,...
i.e. the prime numbers together with their corresponding negative numbers.
As every integer greater than 1 can be written as a product of prime numbers and every integer less than -1 too when we involve the negative versions, it follows that this is an example.

The set of all real numbers is also an example, but a less interesting one, since every element except 0 is a unit!

Last fiddled with by Nick on 2016-04-08 at 08:01 Reason: Typo
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Old 2016-04-08, 08:21   #3
wildrabbitt
 
Jul 2014

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Big thanks.
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