Left Ideal right ideal identity
$begingroup$
If the left ideal equals the right ideal (denoted as $I$) in a ring $R$ does that mean $rI = Ir = I$ for any $r$ inside $R$?
Also, in general, if left ideal is the same as right ideal in a ring, does that mean the ring is commutative? (I think that if the ring is commutative, then any right ideal is the same as left ideal)
abstract-algebra ring-theory
edited Dec 15 '18 at 23:41
AnyAD
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
$endgroup$
add a comment |
$begingroup$
If the left ideal equals the right ideal (denoted as $I$) in a ring $R$ does that mean $rI = Ir = I$ for any $r$ inside $R$?
Also, in general, if left ideal is the same as right ideal in a ring, does that mean the ring is commutative? (I think that if the ring is commutative, then any right ideal is the same as left ideal)
abstract-algebra ring-theory
edited Dec 15 '18 at 23:41
AnyAD
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
$endgroup$
2
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07
add a comment |
$begingroup$
If the left ideal equals the right ideal (denoted as $I$) in a ring $R$ does that mean $rI = Ir = I$ for any $r$ inside $R$?
Also, in general, if left ideal is the same as right ideal in a ring, does that mean the ring is commutative? (I think that if the ring is commutative, then any right ideal is the same as left ideal)
abstract-algebra ring-theory
edited Dec 15 '18 at 23:41
AnyAD
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
$endgroup$
If the left ideal equals the right ideal (denoted as $I$) in a ring $R$ does that mean $rI = Ir = I$ for any $r$ inside $R$?
Also, in general, if left ideal is the same as right ideal in a ring, does that mean the ring is commutative? (I think that if the ring is commutative, then any right ideal is the same as left ideal)
abstract-algebra ring-theory
abstract-algebra ring-theory
edited Dec 15 '18 at 23:41
AnyAD
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
edited Dec 15 '18 at 23:41
AnyAD
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
edited Dec 15 '18 at 23:41
AnyAD
2,098812
edited Dec 15 '18 at 23:41
AnyAD
2,098812
edited Dec 15 '18 at 23:41
AnyAD
2,098812
2,098812
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
asked Dec 15 '18 at 23:32
Extra LearnExtra Learn
113
113
2
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07
add a comment |
2
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07
2
2
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A ring is called right duo If all of its right ideals are also left ideals.
There exist one-sided duo rings
A right-and-left duo ring does not have to be commutative. Example: the quaternions.
if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?
No, it only means $rIsubseteq I$ and $Irsubseteq I$. As mentioned in the comments, any nonzero ideal of $mathbb Z$ will be an example for you.
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
$endgroup$
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
add a comment |
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$begingroup$
A ring is called right duo If all of its right ideals are also left ideals.
There exist one-sided duo rings
A right-and-left duo ring does not have to be commutative. Example: the quaternions.
if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?
No, it only means $rIsubseteq I$ and $Irsubseteq I$. As mentioned in the comments, any nonzero ideal of $mathbb Z$ will be an example for you.
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
$endgroup$
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
add a comment |
$begingroup$
A ring is called right duo If all of its right ideals are also left ideals.
There exist one-sided duo rings
A right-and-left duo ring does not have to be commutative. Example: the quaternions.
if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?
No, it only means $rIsubseteq I$ and $Irsubseteq I$. As mentioned in the comments, any nonzero ideal of $mathbb Z$ will be an example for you.
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
$endgroup$
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
add a comment |
$begingroup$
A ring is called right duo If all of its right ideals are also left ideals.
There exist one-sided duo rings
A right-and-left duo ring does not have to be commutative. Example: the quaternions.
if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?
No, it only means $rIsubseteq I$ and $Irsubseteq I$. As mentioned in the comments, any nonzero ideal of $mathbb Z$ will be an example for you.
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
$endgroup$
A ring is called right duo If all of its right ideals are also left ideals.
There exist one-sided duo rings
A right-and-left duo ring does not have to be commutative. Example: the quaternions.
if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?
No, it only means $rIsubseteq I$ and $Irsubseteq I$. As mentioned in the comments, any nonzero ideal of $mathbb Z$ will be an example for you.
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
edited Dec 16 '18 at 3:45
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
answered Dec 15 '18 at 23:50
rschwiebrschwieb
107k12102251
107k12102251
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
add a comment |
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
$begingroup$
I mean "if I is both a left ideal and a right ideal, does that mean Ir = rI = I for any r inside R?" Sorry for the confusion
$endgroup$
– Extra Learn
Dec 16 '18 at 2:56
add a comment |
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2
$begingroup$
What do you mean “the left ideal equals the right ideal”?
$endgroup$
– rschwieb
Dec 15 '18 at 23:48
$begingroup$
$rI=I$ holds only if $r$ is a unit.This is an example $mathbb{Z}$ is an ideal in itself. However $nmathbb{Z} neq mathbb{Z}$ for any $nneq pm 1$.
$endgroup$
– mouthetics
Dec 16 '18 at 0:02
$begingroup$
@youssefmousaaid that is false: $r$ need not be a unit. For example if e is an idempotent in a commutative ring, $e(eR)$= eR$ , and we can find nonunit idempotents.
$endgroup$
– rschwieb
Dec 16 '18 at 0:54
$begingroup$
@rschwieb Okey!Thanks for the correction.
$endgroup$
– mouthetics
Dec 16 '18 at 1:07