I am currently studying relations between several kinds of rings and domains. I have seen some properties concerning the sum of ideals : when I and J are finitely generated, then I+J is always finitely generated; when I and J are principal, I+J could be not principal (I assume the "easiest" counter-example is given by the ideals $\langle 2 \rangle$ and $\langle X \rangle$ in $\mathbb{Z}[X]$) and the case where the property holds defines the set of Bezout rings. So I was wondering what can be said for intersection instead of sum, more precisely when the intersection of finitely generated (principal) ideals is finitely generated (principal) ? I have read somewhere (don't remember...) that this is related to gcd domains but I cannot find by myself the exact relation. Any comment/help would be welcomed, thanks
2026-04-01 18:36:32.1775068592
Intersection of ideals
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In a domain, the intersection of two finitely generated ideals remains finitely generated iff the domain is coherent, which means that solutions of linear systems can be generated in familiar form, i.e. given a matrix $A$ over the domain there is a matrix $B$ such that
$$ A X = 0 \iff \exists Y\!:\ X = B Y$$
For further details see this talk by T. Coquand, which emphasizes a constructive viewpoint, i.e. coherent rings viewed as constructive approximations of Noetherian rings.