How is a dimension of an ideal defined in $R$
$begingroup$
Suppose I consider the ring $R=Bbb Z_2[x]/langle x^9+1rangle$
$x^9+1$ has a complete factorization into irreducible polynomials like
$x^9+1=(x+1)(x^2+x+1)(x^6+x^3+1)$
Now $I=langle x+1rangle /langle x^9+1rangle $ is an ideal in $R$
I got a question
What is the dimension of $I$ in $R$?
Can someone please tell me how is a dimension of an ideal defined in $R$ and what will be the answer?
abstract-algebra ring-theory
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
$endgroup$
|
show 2 more comments
$begingroup$
Suppose I consider the ring $R=Bbb Z_2[x]/langle x^9+1rangle$
$x^9+1$ has a complete factorization into irreducible polynomials like
$x^9+1=(x+1)(x^2+x+1)(x^6+x^3+1)$
Now $I=langle x+1rangle /langle x^9+1rangle $ is an ideal in $R$
I got a question
What is the dimension of $I$ in $R$?
Can someone please tell me how is a dimension of an ideal defined in $R$ and what will be the answer?
abstract-algebra ring-theory
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
$endgroup$
$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23
|
show 2 more comments
$begingroup$
Suppose I consider the ring $R=Bbb Z_2[x]/langle x^9+1rangle$
$x^9+1$ has a complete factorization into irreducible polynomials like
$x^9+1=(x+1)(x^2+x+1)(x^6+x^3+1)$
Now $I=langle x+1rangle /langle x^9+1rangle $ is an ideal in $R$
I got a question
What is the dimension of $I$ in $R$?
Can someone please tell me how is a dimension of an ideal defined in $R$ and what will be the answer?
abstract-algebra ring-theory
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
$endgroup$
Suppose I consider the ring $R=Bbb Z_2[x]/langle x^9+1rangle$
$x^9+1$ has a complete factorization into irreducible polynomials like
$x^9+1=(x+1)(x^2+x+1)(x^6+x^3+1)$
Now $I=langle x+1rangle /langle x^9+1rangle $ is an ideal in $R$
I got a question
What is the dimension of $I$ in $R$?
Can someone please tell me how is a dimension of an ideal defined in $R$ and what will be the answer?
abstract-algebra ring-theory
abstract-algebra ring-theory
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
asked Nov 25 '18 at 13:45
Join_PhDJoin_PhD
3428
3428
$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23
|
show 2 more comments
$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23
$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
A naïve interpretation of dimension in this case would be the $mathbb Z_2$ dimension. In your case the original ring $R$ is $9$ dimensional over $mathbb Z_2$, and $I$ has codimension $1$ in $R$ (since $(x-1)$ has codimension $1$ in $mathbb Z_2[x]$.) So in that case it would be $8$ dimensional.
But dimension could also refer to the Krull dimension of $R/I$. That seems rather dull though since proper quotients of a polynomial ring over a field are all Artinian, so that the Krull dimension is always $0$.
None of it seems related to the factorization you gave, and without more context it's hard to guess what was intended.
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
$endgroup$
add a comment |
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1 Answer
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$begingroup$
A naïve interpretation of dimension in this case would be the $mathbb Z_2$ dimension. In your case the original ring $R$ is $9$ dimensional over $mathbb Z_2$, and $I$ has codimension $1$ in $R$ (since $(x-1)$ has codimension $1$ in $mathbb Z_2[x]$.) So in that case it would be $8$ dimensional.
But dimension could also refer to the Krull dimension of $R/I$. That seems rather dull though since proper quotients of a polynomial ring over a field are all Artinian, so that the Krull dimension is always $0$.
None of it seems related to the factorization you gave, and without more context it's hard to guess what was intended.
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
$endgroup$
add a comment |
$begingroup$
A naïve interpretation of dimension in this case would be the $mathbb Z_2$ dimension. In your case the original ring $R$ is $9$ dimensional over $mathbb Z_2$, and $I$ has codimension $1$ in $R$ (since $(x-1)$ has codimension $1$ in $mathbb Z_2[x]$.) So in that case it would be $8$ dimensional.
But dimension could also refer to the Krull dimension of $R/I$. That seems rather dull though since proper quotients of a polynomial ring over a field are all Artinian, so that the Krull dimension is always $0$.
None of it seems related to the factorization you gave, and without more context it's hard to guess what was intended.
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
$endgroup$
add a comment |
$begingroup$
A naïve interpretation of dimension in this case would be the $mathbb Z_2$ dimension. In your case the original ring $R$ is $9$ dimensional over $mathbb Z_2$, and $I$ has codimension $1$ in $R$ (since $(x-1)$ has codimension $1$ in $mathbb Z_2[x]$.) So in that case it would be $8$ dimensional.
But dimension could also refer to the Krull dimension of $R/I$. That seems rather dull though since proper quotients of a polynomial ring over a field are all Artinian, so that the Krull dimension is always $0$.
None of it seems related to the factorization you gave, and without more context it's hard to guess what was intended.
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
$endgroup$
A naïve interpretation of dimension in this case would be the $mathbb Z_2$ dimension. In your case the original ring $R$ is $9$ dimensional over $mathbb Z_2$, and $I$ has codimension $1$ in $R$ (since $(x-1)$ has codimension $1$ in $mathbb Z_2[x]$.) So in that case it would be $8$ dimensional.
But dimension could also refer to the Krull dimension of $R/I$. That seems rather dull though since proper quotients of a polynomial ring over a field are all Artinian, so that the Krull dimension is always $0$.
None of it seems related to the factorization you gave, and without more context it's hard to guess what was intended.
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
answered Nov 26 '18 at 13:49
rschwiebrschwieb
105k12101246
105k12101246
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$begingroup$
You’re asking us to guess how dimension is defined in your learning material in this context. At the very least, you should say what the learning material is to help with the guessing.
$endgroup$
– rschwieb
Nov 25 '18 at 13:59
$begingroup$
@rschwieb;Its a question in abstract algebra in our exam,and this the exact question provided,I dont know what it is so I asked if there is some standard notation for this
$endgroup$
– Join_PhD
Nov 25 '18 at 14:12
$begingroup$
you should no tbe asking us question off your exam. You should be asking your teacher.
$endgroup$
– rschwieb
Nov 25 '18 at 14:20
$begingroup$
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
$endgroup$
– Join_PhD
Nov 25 '18 at 14:22
$begingroup$
@rschwieb;Anyway do you have any answer for the question,because I think what we are talking is not going to help either of us
$endgroup$
– Join_PhD
Nov 25 '18 at 14:23