I have the following question:


Let $K$ be an algebraic number field and $[K:Q]=n$. Let $O\_K$ be a full ring of integers of $K$.
Assume that $O \subset O\_K$ is a subring such that rank of $O$ over $Z$ is $n$ ($=\operatorname{rk}\_{Z}O\_K$).
Let $pO\_K=P\_1^{e\_1} ... P\_m^{e\_m}$ is a decomposition into prime ideals in $O\_K$.
(We know that there exists a primary decomposition of ideal $pO$)


Do we know, how many primary ideals in the decomposition are?
Is this number $m$?



Number of ideals in primary decomposition
You don't say which ideal in $O$ you are trying to decompose. I will guess you mean to decompose the ideal $pO$. The number of primary ideals in a *minimal* primary ideal decomposition of $pO$ in $O$ *need not* be the same as the number of prime ideal factors of $pO\_K$ in $O\_K$.


Example: Pick your favorite prime number $p$ and let $O$ be the ring ${\mathbf Z} + pO\_K$. Set ${\mathfrak b} = pO$ and ${\mathfrak P} = pO\_K$. Obviously ${\mathfrak b}$ is an ideal in $O$, but note that ${\mathfrak P}$ is also an ideal in $O$ (it's an ideal in the bigger ring $O\_K$ which happens to lie in the ring $O$, so it's also an ideal in $O$) and ${\mathfrak P}$ is a prime ideal in $O$ (not necessarily in $O\_K$!) since $[O:{\mathfrak P}] = [O:pO\_K] = p$ is a prime. Since, set-theoretically, ${\mathfrak b}$ contains ${\mathfrak P}^2 = p^2O\_K$ and is contained in ${\mathfrak P}$, ${\mathfrak b}$ is a primary ideal, and thus the minimal primary ideal decomposition of ${\mathfrak b} = pO$ in $O$ is ${\mathfrak b}$ itself. I have put no conditions on $p$ at all here (though we chose $O$ depending on $p$) so for instance you could now let $p$ be a prime number which splits completely in $O\_K$ (there are infinitely many of those), and in that case your $m$ is $n$ and that is not the number of ideals in the minimal primary decomposition of ${\mathfrak b} = pO$ in $O$ (well, assuming $K \not= {\mathbf Q}$, so $n > 1$). 


The subtlety here is related to the conductor of your ring $O$. The answer to your question would be $m$ for prime numbers that are relatively prime to the conductor of the ring $O$ you use. In the example above, the conductor of $O$ is $pO\_K$, which contains $p$ and thus one can anticipate complications. 


Perhaps it would help if you explained why you are asking this question.


