Infinite domain with finite number of prime ideals(elements) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:19:23Z http://mathoverflow.net/feeds/question/99606 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99606/infinite-domain-with-finite-number-of-prime-idealselements Infinite domain with finite number of prime ideals(elements) Ostap Chervak 2012-06-14T13:51:34Z 2012-06-15T08:03:23Z <p>While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was the neccesary assumptions, but I failed, since I don't know any "toy"-examples of such rings.</p> <p>I know only one example of this kind ($\mathbf{Q} [x]/ (x^2)$ ) but it's not a domain.</p> <p>So,</p> <p><strong>Are there any infinite domains with finite number of prime ideals?</strong></p> <p><strong>If no, then are there any infinite domains with finite (but nontrivial) number of prime elements?</strong></p> <p>I am interested in noncommutative examples as well. Sorry if this question is too elementary.</p> http://mathoverflow.net/questions/99606/infinite-domain-with-finite-number-of-prime-idealselements/99627#99627 Answer by Ilya Bogdanov for Infinite domain with finite number of prime ideals(elements) Ilya Bogdanov 2012-06-14T17:29:54Z 2012-06-14T17:29:54Z <p>For the commutative case. Take the set of all rational numbers whose denominators are coprime with a fixed integer $n$. Then the only prime ideals are generated by the prime divisors of $n$.</p> <p>More generally, take any PID and take its ring of fractions wrt all the elements coprime with some fixed element; you get a desired ring.</p> http://mathoverflow.net/questions/99606/infinite-domain-with-finite-number-of-prime-idealselements/99681#99681 Answer by Andrew Davies for Infinite domain with finite number of prime ideals(elements) Andrew Davies 2012-06-15T08:03:23Z 2012-06-15T08:03:23Z <p>For a noncommutative example you could take the first Weyl algebra $A_1(k)$ with generated over a field $k$ by elements $x$ and $y$ subject to the relation $xy-yx-1$. This is a Noetherian domain which is simple, that is the only two-sided ideals are 0 and the ring itself.</p>