A property about quasi-primary modules

Question

It is a fact that any discrete valuation domain $R$ has the property "P" that any proper submodule $N$ of any $R$-module $M$ is quasi-primary, in the sense that $\operatorname{rad}(N:M)$ is a prime ideal of $R$, where $(N:M)=\{r\in R : rM⊆N\}$. My conjecture is

Krull dimension of $R$ equal to zero is equivalent to that $R$ has property "P".

It is easy to see that any vector space, or $\mathbb Q$ as a $\mathbb Z$-module has "P".

Is there anybody helping me in this regard? Thanks in advance!

Answer 1

Any valuation domain $R$ satisfies the condition "P".

If $I$ is a proper ideal of a valuation domain $R$, then $\sqrt I$ is a prime ideal.

Let $a,b\in R$ such that $ab\in\sqrt I$. Then there is $n\ge 1$ such that $(ab)^n\in I$. If $a\mid b$ write $b=ax$ and note that $b^{2n}\in I$, so $b\in\sqrt I$.

The conclusion: it is not possible to bound the Krull dimension of rings with the property "P".