Commutative ring with unique primary ideal

Question

I have to prove or give an counter-example to the following

Any commutative ring with unit with a single primary ideal is Noetherian.

I have found the following example: Non-Noetherian ring with a single prime ideal

But in this case the ring has a single prime ideal, i cannot with this assure that this ring has a single primary ideal, but it makes me think that it is possible to construct such a counter-example.

Am i missing something? Can this be proved or is there indeed a counter example?

Answer 1

Let $R$ be a ring with a single primary ideal $P$. Then $P$ is contained in a maximal ideal, which itself is primary, so $P$ is maximal.

Now let $I$ be any proper ideal of $R$. Then the radical of $I$ is the intersection of all prime ideals of $R$ containing $I$, that is $P$. But $P$ is maximal and hence $I$ is primary. Therefore $I$ equals $P$.

So $R$ has a single proper ideal and is in particular Noetherian (even a field).