Example of a ring with no minimal prime ideal
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I am a math student, in the course of abstract algebra we have shown that in a unitary commutative ring every ideal I possesses at least one minimal prime ideal. I am trying to find an example of ideal contents in a non-commutative (not unitary) ring that does not possess a minimal prime ideal.
abstract-algebra ideals maximal-and-prime-ideals
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
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I am a math student, in the course of abstract algebra we have shown that in a unitary commutative ring every ideal I possesses at least one minimal prime ideal. I am trying to find an example of ideal contents in a non-commutative (not unitary) ring that does not possess a minimal prime ideal.
abstract-algebra ideals maximal-and-prime-ideals
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
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add a comment |
$begingroup$
I am a math student, in the course of abstract algebra we have shown that in a unitary commutative ring every ideal I possesses at least one minimal prime ideal. I am trying to find an example of ideal contents in a non-commutative (not unitary) ring that does not possess a minimal prime ideal.
abstract-algebra ideals maximal-and-prime-ideals
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
$endgroup$
I am a math student, in the course of abstract algebra we have shown that in a unitary commutative ring every ideal I possesses at least one minimal prime ideal. I am trying to find an example of ideal contents in a non-commutative (not unitary) ring that does not possess a minimal prime ideal.
abstract-algebra ideals maximal-and-prime-ideals
abstract-algebra ideals maximal-and-prime-ideals
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
edited Dec 15 '18 at 18:20
Antonio B
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
asked Dec 13 '18 at 17:34
Antonio BAntonio B
316
316
add a comment |
add a comment |
1 Answer
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$2mathbb Z/4mathbb Z$ does not have any prime ideals.
If you want an $R$ such that $R^2neq 0$, then $2mathbb Z/8mathbb Z$.
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
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$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
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– Antonio B
Dec 15 '18 at 18:29
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
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– rschwieb
Dec 15 '18 at 19:15
add a comment |
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1 Answer
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1 Answer
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$begingroup$
$2mathbb Z/4mathbb Z$ does not have any prime ideals.
If you want an $R$ such that $R^2neq 0$, then $2mathbb Z/8mathbb Z$.
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
$endgroup$
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
add a comment |
$begingroup$
$2mathbb Z/4mathbb Z$ does not have any prime ideals.
If you want an $R$ such that $R^2neq 0$, then $2mathbb Z/8mathbb Z$.
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
$endgroup$
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
add a comment |
$begingroup$
$2mathbb Z/4mathbb Z$ does not have any prime ideals.
If you want an $R$ such that $R^2neq 0$, then $2mathbb Z/8mathbb Z$.
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
$endgroup$
$2mathbb Z/4mathbb Z$ does not have any prime ideals.
If you want an $R$ such that $R^2neq 0$, then $2mathbb Z/8mathbb Z$.
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
edited Dec 13 '18 at 19:05
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
answered Dec 13 '18 at 18:52
rschwiebrschwieb
106k12102249
106k12102249
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
add a comment |
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
$begingroup$
Hi, do you have an example of a ring that have some prime ideals but there is an ideal that is not contained in any of them?
$endgroup$
– Antonio B
Dec 15 '18 at 18:29
1
1
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
$begingroup$
Take $R$ to be one of the rings above, and use $Rtimes F_2$. Its only prime is $Rtimes {0}$ but ${0}times F$ is not contained in it.
$endgroup$
– rschwieb
Dec 15 '18 at 19:15
add a comment |
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