Left Ideal right ideal identity












2












$begingroup$


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)










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$endgroup$








  • 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
















2












$begingroup$


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)










share|cite|improve this question











$endgroup$








  • 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














2












2








2





$begingroup$


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)










share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 23:41









AnyAD

2,098812




2,098812










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














  • 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










1 Answer
1






active

oldest

votes


















3












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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
















3












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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














3












3








3





$begingroup$

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.






share|cite|improve this answer











$endgroup$



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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 16 '18 at 3:45

























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


















  • $begingroup$
    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


















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