A question about the ideals of a ring of power series.
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Let $F$ a field and we consider the ring of power series $F[[x]]$. After proving that $F[[x]]$ is a P.I.D. and after proving that the ideals of $F[[x]]$ are of the form $(x^k)$ for $kge0$, why $F[x]supset (x)supset(x^2)$? Why $F[[x]]ne (x)$ and $(x)ne (x^2)$.
Thanks!
abstract-algebra proof-verification proof-explanation
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add a comment |
$begingroup$
Let $F$ a field and we consider the ring of power series $F[[x]]$. After proving that $F[[x]]$ is a P.I.D. and after proving that the ideals of $F[[x]]$ are of the form $(x^k)$ for $kge0$, why $F[x]supset (x)supset(x^2)$? Why $F[[x]]ne (x)$ and $(x)ne (x^2)$.
Thanks!
abstract-algebra proof-verification proof-explanation
$endgroup$
2
$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05
add a comment |
$begingroup$
Let $F$ a field and we consider the ring of power series $F[[x]]$. After proving that $F[[x]]$ is a P.I.D. and after proving that the ideals of $F[[x]]$ are of the form $(x^k)$ for $kge0$, why $F[x]supset (x)supset(x^2)$? Why $F[[x]]ne (x)$ and $(x)ne (x^2)$.
Thanks!
abstract-algebra proof-verification proof-explanation
$endgroup$
Let $F$ a field and we consider the ring of power series $F[[x]]$. After proving that $F[[x]]$ is a P.I.D. and after proving that the ideals of $F[[x]]$ are of the form $(x^k)$ for $kge0$, why $F[x]supset (x)supset(x^2)$? Why $F[[x]]ne (x)$ and $(x)ne (x^2)$.
Thanks!
abstract-algebra proof-verification proof-explanation
abstract-algebra proof-verification proof-explanation
asked Dec 10 '18 at 14:56
Jack J.Jack J.
4532419
4532419
2
$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05
add a comment |
2
$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05
2
2
$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05
$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05
add a comment |
1 Answer
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$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.
$endgroup$
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
add a comment |
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1 Answer
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$begingroup$
$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.
$endgroup$
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
add a comment |
$begingroup$
$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.
$endgroup$
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
add a comment |
$begingroup$
$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.
$endgroup$
$(x)$ is the ideal consisting of series with no constant term. Certainly there exist series with nonzero constant term, hence the inclusion is proper. Similarly, $(x^2)$ is the ideal of series with no constant or linear term. Since there exist series with constant terms, or with no constant term but a linear term, the inclusion is proper.
answered Dec 10 '18 at 15:03
Matt SamuelMatt Samuel
38.1k63767
38.1k63767
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
add a comment |
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
$begingroup$
Thanks for your answer. So, if it were $F[[x]]=(x)$ we would be asserting that in $F[[x]]$ every formal power series has costant term null, but this is absurd, because just take $1=1+0x+0x^2+cdots$. Moreover, if it were $(x)=(x^2)$ we would asserting that in $(x)$ every formal series has linear term null, but this is absurd, in fact just take $x=0+1x+0x^2cdots$. It's correct?
$endgroup$
– Jack J.
Dec 10 '18 at 15:23
1
1
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
$begingroup$
@Jack Sure , that works.
$endgroup$
– Matt Samuel
Dec 10 '18 at 15:45
add a comment |
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$begingroup$
In general, for any integral domain, and any nonzero nonunit $x$, $(x^{n+1})subsetneq (x^n)$. For if not, $x^n=x^{n+1}r$ would imply $xr=1$ after cancellation, contradicting that $x$ isn't a unit. But I think Matt's raw description of what each of the ideals $(x^n)$ is like makes it patently clear already for this case.
$endgroup$
– rschwieb
Dec 10 '18 at 15:05