Ideals in valuation rings
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How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?
abstract-algebra ideals maximal-and-prime-ideals valuation-theory
asked Nov 17 at 20:34
Gentiana
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How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?
abstract-algebra ideals maximal-and-prime-ideals valuation-theory
asked Nov 17 at 20:34
Gentiana
254
3
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
1
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?
abstract-algebra ideals maximal-and-prime-ideals valuation-theory
asked Nov 17 at 20:34
Gentiana
254
How do I prove the fact that for any valuation ring $V$ the ideals are totally ordered under inclusion?
abstract-algebra ideals maximal-and-prime-ideals valuation-theory
abstract-algebra ideals maximal-and-prime-ideals valuation-theory
asked Nov 17 at 20:34
Gentiana
254
asked Nov 17 at 20:34
Gentiana
254
edited Nov 17 at 21:02
asked Nov 17 at 20:34
Gentiana
254
asked Nov 17 at 20:34
Gentiana
254
asked Nov 17 at 20:34
Gentiana
254
254
3
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
1
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56
add a comment |
3
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
1
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56
3
3
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
1
1
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56
add a comment |
1 Answer
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Hint:
- Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,bin V$. Show that if $Vanotsubset Vb$, then $Vbsubset Va$.
- Deduce that, if $mathfrak a$ and $mathfrak b$ are two ideals in $V$, if $mathfrak anotsubset mathfrak b$, then $mathfrak bsubset mathfrak a$ (take $ainmathfrak a$, $;anotinmathfrak b$. Show that, for any $binmathfrak b$, $bin Va$).
answered Nov 17 at 21:15
Bernard
116k637108
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint:
- Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,bin V$. Show that if $Vanotsubset Vb$, then $Vbsubset Va$.
- Deduce that, if $mathfrak a$ and $mathfrak b$ are two ideals in $V$, if $mathfrak anotsubset mathfrak b$, then $mathfrak bsubset mathfrak a$ (take $ainmathfrak a$, $;anotinmathfrak b$. Show that, for any $binmathfrak b$, $bin Va$).
answered Nov 17 at 21:15
Bernard
116k637108
add a comment |
up vote
1
down vote
Hint:
- Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,bin V$. Show that if $Vanotsubset Vb$, then $Vbsubset Va$.
- Deduce that, if $mathfrak a$ and $mathfrak b$ are two ideals in $V$, if $mathfrak anotsubset mathfrak b$, then $mathfrak bsubset mathfrak a$ (take $ainmathfrak a$, $;anotinmathfrak b$. Show that, for any $binmathfrak b$, $bin Va$).
answered Nov 17 at 21:15
Bernard
116k637108
add a comment |
up vote
1
down vote
up vote
1
down vote
Hint:
- Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,bin V$. Show that if $Vanotsubset Vb$, then $Vbsubset Va$.
- Deduce that, if $mathfrak a$ and $mathfrak b$ are two ideals in $V$, if $mathfrak anotsubset mathfrak b$, then $mathfrak bsubset mathfrak a$ (take $ainmathfrak a$, $;anotinmathfrak b$. Show that, for any $binmathfrak b$, $bin Va$).
answered Nov 17 at 21:15
Bernard
116k637108
Hint:
- Prove first the principal ideal in $V$ are totally ordered by inclusion: for this, let $a,bin V$. Show that if $Vanotsubset Vb$, then $Vbsubset Va$.
- Deduce that, if $mathfrak a$ and $mathfrak b$ are two ideals in $V$, if $mathfrak anotsubset mathfrak b$, then $mathfrak bsubset mathfrak a$ (take $ainmathfrak a$, $;anotinmathfrak b$. Show that, for any $binmathfrak b$, $bin Va$).
answered Nov 17 at 21:15
Bernard
116k637108
edited Nov 17 at 21:30
answered Nov 17 at 21:15
Bernard
116k637108
answered Nov 17 at 21:15
Bernard
116k637108
answered Nov 17 at 21:15
Bernard
116k637108
116k637108
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add a comment |
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3
What is your definition of a valuation ring?
– Bernard
Nov 17 at 20:55
1
@Bernard That V is a valuation ring if either x or x^-1 is contained in V where x is in its field of fractions F
– Gentiana
Nov 17 at 20:56