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.
abstract-algebra examples-counterexamples maximal-and-prime-ideals
asked Jan 10 at 18:36
Antonio BAntonio B
316
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add a comment |
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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.
abstract-algebra examples-counterexamples maximal-and-prime-ideals
asked Jan 10 at 18:36
Antonio BAntonio B
316
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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
add a comment |
$begingroup$
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.
abstract-algebra examples-counterexamples maximal-and-prime-ideals
asked Jan 10 at 18:36
Antonio BAntonio B
316
$endgroup$
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
abstract-algebra examples-counterexamples maximal-and-prime-ideals
asked Jan 10 at 18:36
Antonio BAntonio B
316
asked Jan 10 at 18:36
Antonio BAntonio B
316
asked Jan 10 at 18:36
Antonio BAntonio B
316
asked Jan 10 at 18:36
Antonio BAntonio B
316
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
add a comment |
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
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint:
$pcdot pinlangle p^2rangle$
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
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add a comment |
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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:
Any patterns present themselves? You might try proving a conjecture...
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
$pcdot pinlangle p^2rangle$
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
$endgroup$
add a comment |
$begingroup$
Hint:
$pcdot pinlangle p^2rangle$
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
$endgroup$
add a comment |
$begingroup$
Hint:
$pcdot pinlangle p^2rangle$
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
$endgroup$
Hint:
$pcdot pinlangle p^2rangle$
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
answered Jan 10 at 18:41
ajotatxeajotatxe
53.9k24090
53.9k24090
add a comment |
add a comment |
$begingroup$
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:
Any patterns present themselves? You might try proving a conjecture...
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
$endgroup$
add a comment |
$begingroup$
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:
Any patterns present themselves? You might try proving a conjecture...
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
$endgroup$
add a comment |
$begingroup$
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:
Any patterns present themselves? You might try proving a conjecture...
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
$endgroup$
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:
Any patterns present themselves? You might try proving a conjecture...
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
answered Jan 10 at 18:42
rschwiebrschwieb
107k12102251
107k12102251
add a comment |
add a comment |
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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