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 $R$?