I asked [this question](https://math.stackexchange.com/q/3062641/660) on Mathematics Stack Exchange but got no answer.


Here is the question:


Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $b=0$).


Assume that $A$ has the following property:


If $\mathfrak p\_1,\dots,\mathfrak p\_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $\mathbb N^k$, then we have 
$$
\mathfrak p\_1^{m\_1}\cdots\mathfrak p\_k^{m\_k}\ne\mathfrak p\_1^{n\_1}\cdots\mathfrak p\_k^{n\_k}.
$$


Does it follow that $A$ is locally noetherian? 


Here are a couple of comments:


In [this answer](https://math.stackexchange.com/a/3062420/660) Badam Baplan pointed out that locally noetherian domains do have the indicated property, and that that some *non-noetherian* domains are *locally* noetherian (the first example seems to have been given by N. Nakano in 1953).


Previously user26857 had observed that [noetherian domains have this property](https://math.stackexchange.com/a/3056360/660), and Julian Rosen had shown that many non-noetherian domains [do *not* have the property](https://math.stackexchange.com/a/3060152/660).



Are unique prime ideal factorization domains locally noetherian?
No, this property does not imply that $A$ is locally Noetherian.


For example, let $F\subset L$ be an extension of fields, and let $A=F+XL[[X]]$ (that is, $A$ is the set of power series over $L$ whose constant term belongs to $F$). Then, $A$ is a one-dimensional local domain with maximal ideal $\mathfrak{m}=XL[[X]]$. Hence, any product of nonzero prime ideals is equal to $\mathfrak{m}^n$ for some integer $n$, and these are all distinct since $\mathfrak{m}^n=X^nL[[X]]$, so your property holds.


However, $A$ is not necessarily Noetherian: indeed, if $\mathfrak{m}$ is finitely generated (as an ideal over $A$) then $L$ will have a finite generating set as an $F$-vector space. In particular, if $[L:F]=\infty$ then $A$ is not Noetherian (and so, being local, it is neither locally Noetherian).


