Let $R$ be a commutative, Noetherian ring with unity. I know that the following is true:
For any ideal $I\subset R$, there are prime ideals $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ such that $\mathfrak{p}_1\cdots\mathfrak{p}_n\subset I$.
If we only consider proper ideals, can we always find prime ideals $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ such that $\mathfrak{p}_1\cdots\mathfrak{p}_n=I$?
I came across this problem on an old qualifying exam and it has me puzzled. I'm beginning to think the problem is stated incorrectly. Are there any counterexamples, or is the result true?
Best Answer
For example in the ring $R=k[x,y]$, the ideal $I=(x,y^2)$ is $P$-primary, where $P = (x, y)$, but is not a power of $P$, so the answer to your question is negative.