Example of a primary ideal that it's not prime












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in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab in I$ and $anotin I$ implies that $ exists n in mathbb{Z}$ such that $b^n in I$. It is evident that prime implies primary, I'm looking for an example that shows that the opposite is not true.










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  • 2




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    Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
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    – Wojowu
    Jan 10 at 18:39










  • $begingroup$
    See the second line in the wikipedia article. At this site, see this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 10 at 19:30


















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$begingroup$


in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab in I$ and $anotin I$ implies that $ exists n in mathbb{Z}$ such that $b^n in I$. It is evident that prime implies primary, I'm looking for an example that shows that the opposite is not true.










share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
    $endgroup$
    – Wojowu
    Jan 10 at 18:39










  • $begingroup$
    See the second line in the wikipedia article. At this site, see this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 10 at 19:30
















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$begingroup$


in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab in I$ and $anotin I$ implies that $ exists n in mathbb{Z}$ such that $b^n in I$. It is evident that prime implies primary, I'm looking for an example that shows that the opposite is not true.










share|cite|improve this question









$endgroup$




in the course of Algebra I studied the primary ideals, an ideals $I$ of a commutative ring with identy is called primary if $ab in I$ and $anotin I$ implies that $ exists n in mathbb{Z}$ such that $b^n in I$. It is evident that prime implies primary, I'm looking for an example that shows that the opposite is not true.







abstract-algebra examples-counterexamples maximal-and-prime-ideals






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asked Jan 10 at 18:36









Antonio BAntonio B

316




316








  • 2




    $begingroup$
    Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
    $endgroup$
    – Wojowu
    Jan 10 at 18:39










  • $begingroup$
    See the second line in the wikipedia article. At this site, see this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 10 at 19:30
















  • 2




    $begingroup$
    Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
    $endgroup$
    – Wojowu
    Jan 10 at 18:39










  • $begingroup$
    See the second line in the wikipedia article. At this site, see this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 10 at 19:30










2




2




$begingroup$
Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
$endgroup$
– Wojowu
Jan 10 at 18:39




$begingroup$
Take your favorite ring (which of course is $mathbb Z$) and try out ideals in it.
$endgroup$
– Wojowu
Jan 10 at 18:39












$begingroup$
See the second line in the wikipedia article. At this site, see this duplicate.
$endgroup$
– Dietrich Burde
Jan 10 at 19:30






$begingroup$
See the second line in the wikipedia article. At this site, see this duplicate.
$endgroup$
– Dietrich Burde
Jan 10 at 19:30












2 Answers
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$begingroup$

Hint:



$pcdot pinlangle p^2rangle$






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    0












    $begingroup$

    This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:



    enter image description here



    Any patterns present themselves? You might try proving a conjecture...






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      2 Answers
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      2 Answers
      2






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      2












      $begingroup$

      Hint:



      $pcdot pinlangle p^2rangle$






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        Hint:



        $pcdot pinlangle p^2rangle$






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          Hint:



          $pcdot pinlangle p^2rangle$






          share|cite|improve this answer









          $endgroup$



          Hint:



          $pcdot pinlangle p^2rangle$







          share|cite|improve this answer












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          answered Jan 10 at 18:41









          ajotatxeajotatxe

          53.9k24090




          53.9k24090























              0












              $begingroup$

              This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:



              enter image description here



              Any patterns present themselves? You might try proving a conjecture...






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:



                enter image description here



                Any patterns present themselves? You might try proving a conjecture...






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:



                  enter image description here



                  Any patterns present themselves? You might try proving a conjecture...






                  share|cite|improve this answer









                  $endgroup$



                  This is the graphic at wikipedia's prime ideal page currently. I thought it also appeared on the primary ideal page, but it looks like it doesn't:



                  enter image description here



                  Any patterns present themselves? You might try proving a conjecture...







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 10 at 18:42









                  rschwiebrschwieb

                  107k12102251




                  107k12102251






























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