Suppose $R$ is a Noetherian ring and $P$ is a prime ideal. If the number of generators of $PR_P$ as an ideal in $R_P$ is $n$, can we say anything about the number of generators of $P$ as an ideal of $R$?
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Certainly $P$ has at least $n$ generators, as the images of the generators in $R_p$ generate $PR_p$. As you can see from Andrea's comment, the we cannot give an upper bound in general. This is because the number of generators is not really a "nice" property of ideals. A closely related property is the height of an ideal, which for a prime ideal $P$ is the supremum of the lengths of chains of prime ideals contained in $P$, and for a general ideal is the infimum of the heights of prime ideals containing it. Since we have a bijection between the ideals of $R$ contained in $P$ and the ideals of $R_P$, the height of $P$ is the same as the height of $PR_P$. While this is not quite what you're looking for, it's close. By the generalized principal ideal theorem, the height of an ideal in a Noetherian ring is at most the number of generators. In some cases we can say more, especially for prime ideals. For example, prime ideals of height $1$ in a UFD have exactly one generator. In a regular local ring, the height of the maximal ideal is equal to its number of generators. |
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