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I know that in rings which are not UFDs, the GCD and LCM may not exist. However, I don't know whether it is equivalent the existence of the GCD and the LCM in one of those rings, i.e.

I am looking for an example where a couple of elements have a GCD and fail to have a LCM (or viceversa).

Can anyone help me?

user26857
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J. Karen
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    See here: https://math.stackexchange.com/questions/1449521/gcd-domain-is-lcm-domain The first comment gives a counter-example you are looking for. – Dirk May 15 '17 at 08:27
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    Viceversa isn't possible: if LCM of two elements there exists, then GCD of them there exists, too. The converse of this is not true as shows $X^2$ and $X^3$ in the ring $K[X^2,X^3]$: $\gcd(X^2,X^3)=1$ and LCM doesn't exist. – user26857 May 15 '17 at 21:20
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    @Bemte It seems you wanted to say "the second comment ". – user26857 May 15 '17 at 21:51
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    @user26857 Seems like it, yes. Regarding your example: Why isn't lcm$(X^2,X^3) = X^5$? So why doesn't it exist? – Dirk May 16 '17 at 08:54
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    @Bemte In that case $X^5\mid X^6$ (since $X^6$ is also a common multiple), and this is impossible in our ring (which doesn't contain $X$). – user26857 May 16 '17 at 09:07

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