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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

Robert Cardona
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Nekochan
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  • There are of course some trivial cases in which your question is true (and also where your statement about sums is false.) In a Principal Ideal Domain, every ideal is principal so the sum of two ideals and the intersection of two ideals is also principal. When, you say gcd domain, I think you mean a Euclidean Domain, which is stronger than being a Principal Ideal Domain. Additionally, in a Noetherian ring, every ideal is finitely generated. – Siddharth Venkatesh Jun 03 '14 at 14:07
  • http://math.stackexchange.com/questions/295875/intersection-of-finitely-generated-ideals – user26857 Jun 04 '14 at 19:11

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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.

Bill Dubuque
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