Commutative $mathbb{Q}$-algebras which are not integral domains
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Is it possible to find all commutative $mathbb{Q}$-algebras which are not integral domains?
An example for such algebra is $frac{mathbb{Q}[t]}{(t^2-1)}$.
More generally, $frac{mathbb{Q}[t]}{(h)}$, where $h in Q[t]$ is reducible in $mathbb{Q}[t]$.
Any hints and comments are welcome!
algebraic-geometry ring-theory commutative-algebra
asked Nov 18 at 1:16
user237522
2,0101617
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show 3 more comments
up vote
1
down vote
favorite
1
Is it possible to find all commutative $mathbb{Q}$-algebras which are not integral domains?
An example for such algebra is $frac{mathbb{Q}[t]}{(t^2-1)}$.
More generally, $frac{mathbb{Q}[t]}{(h)}$, where $h in Q[t]$ is reducible in $mathbb{Q}[t]$.
Any hints and comments are welcome!
algebraic-geometry ring-theory commutative-algebra
asked Nov 18 at 1:16
user237522
2,0101617
I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
1
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
1
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24
|
show 3 more comments
up vote
1
down vote
favorite
1
up vote
1
down vote
favorite
1
1
Is it possible to find all commutative $mathbb{Q}$-algebras which are not integral domains?
An example for such algebra is $frac{mathbb{Q}[t]}{(t^2-1)}$.
More generally, $frac{mathbb{Q}[t]}{(h)}$, where $h in Q[t]$ is reducible in $mathbb{Q}[t]$.
Any hints and comments are welcome!
algebraic-geometry ring-theory commutative-algebra
asked Nov 18 at 1:16
user237522
2,0101617
Is it possible to find all commutative $mathbb{Q}$-algebras which are not integral domains?
An example for such algebra is $frac{mathbb{Q}[t]}{(t^2-1)}$.
More generally, $frac{mathbb{Q}[t]}{(h)}$, where $h in Q[t]$ is reducible in $mathbb{Q}[t]$.
Any hints and comments are welcome!
algebraic-geometry ring-theory commutative-algebra
algebraic-geometry ring-theory commutative-algebra
asked Nov 18 at 1:16
user237522
2,0101617
asked Nov 18 at 1:16
user237522
2,0101617
asked Nov 18 at 1:16
user237522
2,0101617
asked Nov 18 at 1:16
user237522
2,0101617
asked Nov 18 at 1:16
user237522
2,0101617
2,0101617
I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
1
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
1
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24
|
show 3 more comments
I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
1
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
1
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24
I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
1
1
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
1
1
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24
|
show 3 more comments
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I think for finitely generated $mathbb{Q}-$algebras we can realize them as quotients of polynomial rings over $mathbb{Q}$ over finitely many variables and the ideal we quotient them over have to be prime and I think we can characterize which ideals are prime in these polynomial rings. Not sure...
– 伽罗瓦
Nov 18 at 1:21
@伽罗瓦, thank you for your comment! It sounds a good idea to first concentrate on f.g. $mathbb{Q}$-algebras. If I am not wrong, it is difficult to find all prime ideals in $mathbb{Q}[x_1,ldots,x_n]$...so there is no easy answer to my question.
– user237522
Nov 18 at 1:30
This random paper I just found might help: ams.org/journals/proc/1997-125-01/S0002-9939-97-03663-0/…
– 伽罗瓦
Nov 18 at 1:32
1
Why do you want to find all such algebras? What description is good enough for you? It's trivial to say that any such fg algebra is the quotient of a polynomial ring by a non-prime ideal, for example: do you want a list of all non-prime ideals?
– KReiser
Nov 18 at 1:34
1
There are a countably infinite collection of non-prime ideals for any $Bbb Q[x_1,cdots,x_n]$: this is a lot! Algorithms exist to determine whether an ideal is prime or not - see Macaulay2's isPrime function, which works via elimination theory and Grobner bases. So nobody's going to be able to write down the whole list for you in any meaningful fashion, just nobody can tell you an explicit list of all composite integers.
– KReiser
Nov 18 at 2:24