How is a dimension of an ideal defined in $R$
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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 at 13:45
Join_PhD
827
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show 2 more comments
up vote
0
down vote
favorite
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 at 13:45
Join_PhD
827
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.
– rschwieb
Nov 25 at 13:59
@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
– Join_PhD
Nov 25 at 14:12
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@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
– Join_PhD
Nov 25 at 14:23
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 13:45
Join_PhD
827
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 at 13:45
Join_PhD
827
asked Nov 25 at 13:45
Join_PhD
827
asked Nov 25 at 13:45
Join_PhD
827
asked Nov 25 at 13:45
Join_PhD
827
asked Nov 25 at 13:45
Join_PhD
827
827
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.
– rschwieb
Nov 25 at 13:59
@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
– Join_PhD
Nov 25 at 14:12
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@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
– Join_PhD
Nov 25 at 14:23
|
show 2 more comments
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.
– rschwieb
Nov 25 at 13:59
@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
– Join_PhD
Nov 25 at 14:12
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@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
– Join_PhD
Nov 25 at 14:23
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.
– rschwieb
Nov 25 at 13:59
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.
– rschwieb
Nov 25 at 13:59
@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
– Join_PhD
Nov 25 at 14:12
@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
– Join_PhD
Nov 25 at 14:12
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@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
– Join_PhD
Nov 25 at 14:23
@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
– Join_PhD
Nov 25 at 14:23
|
show 2 more comments
1 Answer
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votes
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0
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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 at 13:49
rschwieb
104k1299238
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
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 at 13:49
rschwieb
104k1299238
add a comment |
up vote
0
down vote
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 at 13:49
rschwieb
104k1299238
add a comment |
up vote
0
down vote
up vote
0
down vote
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 at 13:49
rschwieb
104k1299238
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 at 13:49
rschwieb
104k1299238
answered Nov 26 at 13:49
rschwieb
104k1299238
answered Nov 26 at 13:49
rschwieb
104k1299238
answered Nov 26 at 13:49
rschwieb
104k1299238
104k1299238
add a comment |
add a comment |
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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.
– rschwieb
Nov 25 at 13:59
@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
– Join_PhD
Nov 25 at 14:12
you should no tbe asking us question off your exam. You should be asking your teacher.
– rschwieb
Nov 25 at 14:20
@rschwieb;It is a PhD qualifying exam,so I have no access to the teacher of the institutes
– Join_PhD
Nov 25 at 14:22
@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
– Join_PhD
Nov 25 at 14:23