Ring with infinitely many totally ordered prime ideals
4
$begingroup$
I was looking for a specific ring with infinitely many prime ideals such that they are totally ordered by inclusion.
A valuation ring with rank $mathbb{N} cup infty$ should work, or something like that, but I don't know any.
abstract-algebra algebraic-geometry ring-theory maximal-and-prime-ideals
asked Nov 27 '18 at 23:10
KatoKato
584
$endgroup$
|
show 1 more comment
4
$begingroup$
I was looking for a specific ring with infinitely many prime ideals such that they are totally ordered by inclusion.
A valuation ring with rank $mathbb{N} cup infty$ should work, or something like that, but I don't know any.
abstract-algebra algebraic-geometry ring-theory maximal-and-prime-ideals
asked Nov 27 '18 at 23:10
KatoKato
584
$endgroup$
1
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26
|
show 1 more comment
4
4
4
1
$begingroup$
I was looking for a specific ring with infinitely many prime ideals such that they are totally ordered by inclusion.
A valuation ring with rank $mathbb{N} cup infty$ should work, or something like that, but I don't know any.
abstract-algebra algebraic-geometry ring-theory maximal-and-prime-ideals
asked Nov 27 '18 at 23:10
KatoKato
584
$endgroup$
I was looking for a specific ring with infinitely many prime ideals such that they are totally ordered by inclusion.
A valuation ring with rank $mathbb{N} cup infty$ should work, or something like that, but I don't know any.
abstract-algebra algebraic-geometry ring-theory maximal-and-prime-ideals
abstract-algebra algebraic-geometry ring-theory maximal-and-prime-ideals
asked Nov 27 '18 at 23:10
KatoKato
584
asked Nov 27 '18 at 23:10
KatoKato
584
edited Nov 28 '18 at 11:07
Kato
asked Nov 27 '18 at 23:10
KatoKato
584
asked Nov 27 '18 at 23:10
KatoKato
584
asked Nov 27 '18 at 23:10
KatoKato
584
584
1
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26
|
show 1 more comment
1
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26
1
1
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26
|
show 1 more comment
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1
$begingroup$
@JeskoHüttenhain Probably the meaning is that the set of prime ideals is totally ordered by inclusion.
$endgroup$
– egreg
Nov 27 '18 at 23:33
$begingroup$
@egreg that makes perfect sense, thanks for helping me out there.
$endgroup$
– Jesko Hüttenhain
Nov 27 '18 at 23:34
$begingroup$
Do you mean that there is some set of prime ideals which is totally ordered by inclusion or that the collection of all prime ideals needs to be totally ordered by inclusion? Also, $mathbb{N} cup { infty }$ isn't a group under addition.
$endgroup$
– Qiaochu Yuan
Nov 28 '18 at 8:45
$begingroup$
Thanks, edited, I mean all primes.
$endgroup$
– Kato
Nov 28 '18 at 11:08
$begingroup$
@JyrkiLahtonen But there are other prime ideals.
$endgroup$
– egreg
Nov 28 '18 at 11:26