Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j > i$. We consider the ring $R_{\infty}$ defined by

$R_{\infty} \colon= \underset{i \geq 1}\varprojlim \,R_i$.

$R_{\infty}$ might not have finite Krull dimension, nor be noetherian. Choose finitely many non-zero elements $r_1,...,r_{n} \in R_{\infty}$ and consider the ideal ${\frak a} \colon= (r_1,...,r_{n})$ of $R_{\infty}$.

## Q1. Does a prime ideal ${\frak P}$ of $R_{\infty}$ always exist such that ${\mathrm{ht}}({\frak P}) < \infty$ and ${\frak P} \supset {\frak a}$ ?

Next let us localize $R_{\infty}$ at a prime ${\frak Q}$ of $R_{\infty}$ such that ${\mathrm{ht}(\frak Q)} < \infty$ and we denote it by $R_{\infty, {\frak Q}}$.