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LSpice
LSpice
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I$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $rad(I)$$\rad(I)$ is irreducible but not prime ideal.

In case of $R$ is Noetherian,radical the radical of $I$ isbeing irreducible implies $rad(I)$$\rad(I)$ is primary. Then it is straight forward to see that $rad(I)$$\rad(I)$ is prime. So we want to look at non Noetherian rings. But I am not able to find such examples. Also, $I$ cannot be primary as $rad(I)$$\rad(I)$ is prime if $I$ is primary.

I am searching an example of ideal $I$ of a ring $R$ such that $rad(I)$ is irreducible but not prime ideal.

In case of $R$ is Noetherian,radical of $I$ is irreducible implies $rad(I)$ is primary. Then it is straight forward to see that $rad(I)$ is prime. So we want to look at non Noetherian rings. But I am not able to find such examples. Also, $I$ cannot be primary as $rad(I)$ is prime if $I$ is primary.

$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal.

In case $R$ is Noetherian, the radical of $I$ being irreducible implies $\rad(I)$ is primary. Then it is straight forward to see that $\rad(I)$ is prime. So we want to look at non Noetherian rings. But I am not able to find such examples. Also, $I$ cannot be primary as $\rad(I)$ is prime if $I$ is primary.

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Melon_Musk
Melon_Musk
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An example of radical ideal which is irreducible but not prime

I am searching an example of ideal $I$ of a ring $R$ such that $rad(I)$ is irreducible but not prime ideal.

In case of $R$ is Noetherian,radical of $I$ is irreducible implies $rad(I)$ is primary. Then it is straight forward to see that $rad(I)$ is prime. So we want to look at non Noetherian rings. But I am not able to find such examples. Also, $I$ cannot be primary as $rad(I)$ is prime if $I$ is primary.