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We're in an integral domain with unity 1 $\neq$ 0. Suppose that the highest common factor between x,y is 1 and the highest common factor for x,z is 1.

Show that $x \mid yz$ implies that $x$ is a unit, or provide a counterexample.

I'm stuck. I don't have that we are in a unique factorization domain, I don't have that this ring is Noetherian.

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Thanks, I wrote the problem wrong :). Just fixed it. – user28877 Apr 14 '12 at 21:40
Did you mean, the highest common factor between $x$ and $y$, and between $x$ and $z$ is $1$? If so, look at $\mathbb{Z}[\sqrt{-5}]$, $x=2$, $y=1+\sqrt{-5}$, and $z=1-\sqrt{-5}$. – Arturo Magidin Apr 14 '12 at 21:40
What does a "highest" common factor mean in a general integral domain (that is, without any assumption that it is Euclidean)? – Henning Makholm Apr 14 '12 at 21:46
(It is probably reasonable to interpret it as "all common factors of $x$ and $y$ are units", or equivalently "the ideal generated by $x$ and $y$ is the entire ring" -- but would be nice to have that confirmed explicitly). – Henning Makholm Apr 14 '12 at 21:52
it means it is a multiple of any other common factor. – user28877 Apr 14 '12 at 21:53
up vote 1 down vote accepted

Hint $\ $ For the special case $\rm\:x\:$ is irreducible, a counterexample would be an irreducible element that is not prime. These are easy to find in non-UFD number rings. See also this post.

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This is interesting. So then I am clearly stuck on a problem. I am asked to find that if $a$ and $a$ have a highest common factor, then they have a least common multiple. So here is my strategy: I let $(a,b)$ be the hcf, write $(a,b)a'=a$ and $(a,b)b'=b$ and claim that my candidate is $(a,b)a'b'$ is my lcm. I will repost this as a new question. – user28877 Apr 14 '12 at 21:55

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