These questions are inspired from the well known fact (by Sally et. al.) as follows:


**Theorem 1.** Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension one. Then the minimal number of generators of ideals of $R$ is bounded above by a constant i.e. there exists a positive integer $C$ such that $\ell (I/\mathfrak{m}I) \leq C$ for all ideal $I$. 


In higher dimension, it is easy to see Theorem 1 is not true.


**Question 2.** Let $(R, \mathfrak{m})$ be a Noetherian local domain of dimension two. Does the exist a positive integer $C$ such that $\ell (\mathfrak{p}/\mathfrak{mp}) \leq C$ for all prime ideal $\mathfrak{p}$ of $R$? 


**Question 3.** Let $(R, \mathfrak{m})$ be a Noetherian local domain of dimension two. Does the exist a positive integer $C$ such that $\ell (\mathfrak{p}R\_{\mathfrak{p}}/\mathfrak{p}^2R\_{\mathfrak{p}}) \leq C$ for all prime ideal $\mathfrak{p}$ of $R$?



uniform bound of the number of generators of prime ideals
Here is answer for question 3 in arbitrary dimension with a certain restriction on $R$.


We assume $R$ is a image of a regular local ring $(S, \mathfrak{n})$ (we always have this assumption by passing the completion). $R = S/ I$ for some ideal $I$ of $S$. Recall that the embedded dimension of $R$ is $\mu(\mathfrak{m}) = \ell (\mathfrak{m}/\mathfrak{m}^2)$, we denote it by $C$. It is not difficult to see that we can assume that the embedded dimension of $S$ is equal to $\mu(\mathfrak{m})$. So $S$ is a regular local ring of dimension $C$. 


Now let $\mathfrak{p}$ be a prime ideal of $R$. Let $\mathfrak{q} \in Spec (S)$ such that $\mathfrak{p} = \mathfrak{q}/I$. It is well known that $S\_{\mathfrak{q}}$ is also a regular local ring of (embedded) dimension $\leq C$. Moreover $R\_{\mathfrak{p}} = S\_{\mathfrak{q}}/I S\_{\mathfrak{q}}$. So the embedded dimension of $R\_{\mathfrak{p}}$ is less than or equal to the embedded dimension of $S\_{\mathfrak{q}}$. Thus $\mu (\mathfrak{p}R\_{\mathfrak{p}}) = \ell (\mathfrak{p}R\_{\mathfrak{p}}/ \mathfrak{p}^2R\_{\mathfrak{p}}) \leq C$.


Edit:


I have just read from Sally's book (page 52) the following result:


Theorem: Let $(R, \mathfrak{m})$ be a NOetherian local ring. Then $\dim R \leq 2$ iff there is a uniform bound of the number of generators of all ideals which do not have $\mathfrak{m}$ as a associated prime.


Thus, Question 3 has an affirmative answer. 


