Talk:GCD domain
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Gauss lemma
[edit]Doesn't the Gauss lemma hold for GCD domains? -- Taku (talk) 03:19, 13 February 2009 (UTC)
- Which Gauss lemma? Product of primitive polys is primitive? Irreducible implies irreducible over field of fractions? Algebraist 03:30, 13 February 2009 (UTC)
- I meant the first one. (I didn't know the second.) -- Taku (talk) 04:19, 13 February 2009 (UTC)
- They're both called Gauss's lemma, according to my lecturer and our article. The first is true for GCD domains, according to the Chapman & Glaz book. The second easily follows from the first, but I don't have a source for that at the moment. Algebraist 16:31, 13 February 2009 (UTC)
- I meant the first one. (I didn't know the second.) -- Taku (talk) 04:19, 13 February 2009 (UTC)
Polynomial ring of GCD domain
[edit]In the reference for the claim that R[X1,...,Xn] is a GCD-domain if R is a GCD-domain, I can find that R[X1,X1-1] is a GCD-domain if R is a GCD-domain, which is another statement. — Preceding unsigned comment added by 77.248.14.251 (talk) 00:32, 30 January 2016 (UTC)
The definition
[edit]I'm not happy with the definition given in this article. Say a,b are elements in a domain. It is not a priori clear, why 'unique minimal principal ideal containing (a,b)' should be the same as 'smallest principal ideal containing (a,b)'. The latter is definitely the correct definition, as it is equivalent to the definition of gcd in terms of elements. (A gcd of a and b is an element g that divides both a and b, such that if x divides a and b, then x divides g.)
(I think there are weird counterexamples if one doesn't add an assumption. There could be an infinite descending chain of principal ideals containing (a,b) (all not minimal), there could be a unique minimal principal ideal containing (a,b) and it is not the smallest if it isn't contained in the infinite chain.) --Wandynsky (talk) 09:34, 14 July 2021 (UTC)
- "Finite intersections of principal ideals are principal" is the best definition for me. A definition involving ideals rather than elements seems more natural, and it serves as some kind of dual to the definition of Bézout domain: finitely generated ideals are principal. (Although it is not a priori clear using this definition that Bézout domains are GCD domains). 129.104.241.34 (talk) 15:56, 30 November 2024 (UTC)
- This is a correct and important observation. I changed “unique minimal” to “minimum” in the article.
- The definition was technically correct in the sense that if R is an integral domain with the property that any two elements are contained in a unique minimal principal ideal, then you can also show that any two elements are contained in a minimum principal ideal, i.e. R is a GCD domain. However, this is not immediately obvious and you can construct an integral domain for which there exist two elements contained in a unique minimal principal ideal but not contained in a minimum principal ideal. Julian Benali (talk) 19:43, 4 June 2026 (UTC)
Every irreducible element of a GCD domain is prime
[edit]So does the kind of integral domains having the property "every irreducible element is prime" has a name? 129.104.241.181 (talk) 11:56, 27 November 2024 (UTC)
If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain
[edit]So why not say "if and only if"? Is the converse too obvious (by definition of the GCD) to be mentioned? 129.104.241.34 (talk) 15:51, 30 November 2024 (UTC)