Show that the center of a division ring is a field.
up vote
1
down vote
favorite
I tried the proof, but wasn't able to proceed further:
let a and b be any elements belonging to Z(center),
Now a.b=b.a
hence Z is commutative.
Now proving that Z is a division ring:(then we can show that since every commutative division ring is field, Z is a field)
NO CLUE...
abstract-algebra ring-theory
asked Nov 15 at 20:50
Cosmic
448
add a comment |
up vote
1
down vote
favorite
I tried the proof, but wasn't able to proceed further:
let a and b be any elements belonging to Z(center),
Now a.b=b.a
hence Z is commutative.
Now proving that Z is a division ring:(then we can show that since every commutative division ring is field, Z is a field)
NO CLUE...
abstract-algebra ring-theory
asked Nov 15 at 20:50
Cosmic
448
2
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
1
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I tried the proof, but wasn't able to proceed further:
let a and b be any elements belonging to Z(center),
Now a.b=b.a
hence Z is commutative.
Now proving that Z is a division ring:(then we can show that since every commutative division ring is field, Z is a field)
NO CLUE...
abstract-algebra ring-theory
asked Nov 15 at 20:50
Cosmic
448
I tried the proof, but wasn't able to proceed further:
let a and b be any elements belonging to Z(center),
Now a.b=b.a
hence Z is commutative.
Now proving that Z is a division ring:(then we can show that since every commutative division ring is field, Z is a field)
NO CLUE...
abstract-algebra ring-theory
abstract-algebra ring-theory
asked Nov 15 at 20:50
Cosmic
448
asked Nov 15 at 20:50
Cosmic
448
asked Nov 15 at 20:50
Cosmic
448
asked Nov 15 at 20:50
Cosmic
448
asked Nov 15 at 20:50
Cosmic
448
448
2
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
1
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08
add a comment |
2
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
1
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08
2
2
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
1
1
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
Hopefully you can already prove:
The center of a ring is a ring, in fact, a commutative ring.
Furthermore, if $D$ is a division ring, then for all $xin Z(D)$, if $xneq 0$, then $x^{-1}$ exists somewhere in $D$.
Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.
So, the task is clear: if $0neq xin Z(D)$, prove $x^{-1}in Z(D)$.
answered Nov 15 at 22:23
rschwieb
103k1299238
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Hopefully you can already prove:
The center of a ring is a ring, in fact, a commutative ring.
Furthermore, if $D$ is a division ring, then for all $xin Z(D)$, if $xneq 0$, then $x^{-1}$ exists somewhere in $D$.
Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.
So, the task is clear: if $0neq xin Z(D)$, prove $x^{-1}in Z(D)$.
answered Nov 15 at 22:23
rschwieb
103k1299238
add a comment |
up vote
3
down vote
accepted
Hopefully you can already prove:
The center of a ring is a ring, in fact, a commutative ring.
Furthermore, if $D$ is a division ring, then for all $xin Z(D)$, if $xneq 0$, then $x^{-1}$ exists somewhere in $D$.
Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.
So, the task is clear: if $0neq xin Z(D)$, prove $x^{-1}in Z(D)$.
answered Nov 15 at 22:23
rschwieb
103k1299238
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Hopefully you can already prove:
The center of a ring is a ring, in fact, a commutative ring.
Furthermore, if $D$ is a division ring, then for all $xin Z(D)$, if $xneq 0$, then $x^{-1}$ exists somewhere in $D$.
Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.
So, the task is clear: if $0neq xin Z(D)$, prove $x^{-1}in Z(D)$.
answered Nov 15 at 22:23
rschwieb
103k1299238
Hopefully you can already prove:
The center of a ring is a ring, in fact, a commutative ring.
Furthermore, if $D$ is a division ring, then for all $xin Z(D)$, if $xneq 0$, then $x^{-1}$ exists somewhere in $D$.
Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.
So, the task is clear: if $0neq xin Z(D)$, prove $x^{-1}in Z(D)$.
answered Nov 15 at 22:23
rschwieb
103k1299238
answered Nov 15 at 22:23
rschwieb
103k1299238
answered Nov 15 at 22:23
rschwieb
103k1299238
answered Nov 15 at 22:23
rschwieb
103k1299238
103k1299238
add a comment |
add a comment |
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2
Take an element in the center and show its inverse is in the center too
– gd1035
Nov 15 at 20:51
I am sorry but, i am not getting how will it help?
– Cosmic
Nov 15 at 20:56
1
Z is a division ring if $zin$ Z $rightarrowexists yin$ Z such that $zy=1$, so you just need to take an arbitrary element in Z and show that its inverse is in Z
– gd1035
Nov 15 at 21:08