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I am working with some non-principal ideals $I=\langle a,b\rangle$, $J=\langle c,d\rangle$ in a nicely behaved Noetherian UFD (the Laurent polynomial ring in finitely many commuting variables with coefficients in $\mathbb{Z}$). As an alternative approach to Gröbner bases, I was very interested in the suggestion that $$I\cap J=\langle \textrm{lcm}(a,c), \textrm{lcm}(a,d), \textrm{lcm}(b,c), \textrm{lcm}(b,d)\rangle$$ (see the previous question: generators of intersection of ideals). I cannot seem to find a suitable reference for this, and I am not able to derive the result directly by myself. I am familiar with the intersection of principal ideals in a UFD being given by the least common multiple, but I cannot deduce this non-principal case. Any reference or explanation would be very nice to see.

Something that seems strange about this claim is that if $\textrm{lcm}(a,c)=ac$, $\textrm{lcm}(a,d)=ad$, $\textrm{lcm}(b,c)=bc$, $\textrm{lcm}(b,d)=bd$, then the result above would show that $I\cap J = IJ$, which can be a delicate question (see this related question).

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For Gröbner bases maybe it's enough to know such a result for monomial ideals (in polynomial rings over a field, of course), and this indeed holds. In turn, I have serious doubts about the @JSchlather's comment to this question. –  user26857 Feb 6 at 21:21
    
Yes, it seems like an almost too good to be true result, in the generality suggested by the comment you refer to. Nonetheless, a counterexample or some further expert elucidation would still be appreciated. –  RMiles Feb 8 at 4:30

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