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Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.

"Easy" means that I can explain it to my undergrad students, although I will be happy with any example.

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5 Answers

up vote 6 down vote accepted

Here's an example stolen blatantly from wikipedia.

Let $R=\mathbb{Z}[\sqrt{-3}]$, let $a=4=2*2=(1+\sqrt{-3})(1-\sqrt{-3})$ and $b=2(1+\sqrt{-3})$. Now, $2$ and $1+\sqrt{-3}$ are both maximal among divisors, but are not associates, thus, there is not GCD.

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More generally, GCD domains are integrally closed. (Note that the real problem here is not about two elements, but one: the lattice of divisors of 4 does not have suprema.) –  Qiaochu Yuan Jan 8 '10 at 5:26
Thanks. I should learn to look up wikipedia before asking. –  Alfonso Gracia-Saz Jan 8 '10 at 5:43
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I should point out, there are plenty of examples in integrally closed rings. For example:

In $k[a,b,c,d]/(ad-bc)$, there is no GCD of $ad$ and $ab$. (Note that $a$ and $b$ are both common divisors.)

In $\mathbb{Z}[\sqrt{-5}]$, there is no GCD of $6$ and $2 (1+\sqrt{-5})$. (Note that $2$ and $1+\sqrt{-5}$ are both common divisors.)

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Since Charles has already given you an example, I'll just mention that there is a name for integral domains in which any two non-zero elements have a gcd: GCD-Domains.

See also Pete's response to “Counter”-example for Gauss’s Lemma on irreducible polynomials.

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I'd actually never heard of GCD-domains and GL-domains until this question, partly because I had missed that thread. Thanks Ben! –  Charles Siegel Jan 8 '10 at 5:37
Right, and in the Noetherian case a GCD-domain is precisely a UFD. So there are lots of examples. –  Pete L. Clark Jan 8 '10 at 6:03
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It deserves to be much better known that nonexistant GCDs (and, similarly, nonprincipal ideals) arise immediately from any failure of Euclid's Lemma, and this provides an illuminating way to view many of the standard examples. Below is a detailed explanation extracted from one of my sci.math.research posts [2]. The results below hold true in any domain D.

LEMMA: (a,b) = (ac,bc)/c if (ac,bc) exists

Proof: d|a,b <=> dc|ac,bc <=> dc|(ac,bc) <=> d|(ac,bc)/c. QED

EUCLID'S LEMMA: a|bc and (a,b)=1 => a|c, if (ac,bc) exists

Proof: a|ac,bc => a|(ac,bc) = (a,b)c = c via Lemma. QED

Therefore if a,b,c fail to satisfy the implication in Euclid's Lemma, namely if (a,b) = 1 and a|bc, not a|c, then one immediately deduces that the gcd (ac,bc) fails to exist in D.

E.g. David Speyer's example above, and Khurana's example in [1] (= Theorem 31 in Pete L. Clark's [0]) are simply specializations where a,b,c = p,1+w,1-w in a quadratic number (sub)ring Z[w], ww = -d.

[0] Clark, Pete. L. Factorization in integral domains. 29pp. 2009. http://math.uga.edu/~pete/factorization.pdf

[1] D. Khurana, On GCD and LCM in domains: A Conjecture of Gauss. Resonance 8 (2003), 72-79. http://www.ias.ac.in/resonance/June2003/pdf/June2003Classroom.pdf

[2] sci.math.research, 3/12/09, seeking comments on expository article on factorization

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I'd be very grateful if the person who downvoted the above post could please explain why - even if anonymously. On other forums I've received many positive responses on variants of this post, so I'm quite puzzled why the reception would be different here. If something is not clear please feel welcome to ask questions and I am more than happy to elaborate. –  Bill Dubuque Jul 12 '10 at 16:40
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Another possible answer, following Qiaochu's comment: The elements $x^2$ and $x^3$ (Edit: $x^5$ and $x^6$) in $k[x^2, x^3]$ (alternatively, $k[x,y]/(x^3-y^2)$), where $k$ is a nonzero ring.

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Non-integrally closed rings make my head spin, but don't those two elements have gcd 1? (I can't find any non-constant common divisors). I think x^4 and x^5 should work here, though. –  Alison Miller Jan 8 '10 at 19:42
You're quite right, thanks. –  S. Carnahan Jan 8 '10 at 19:53
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