The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., x_r]$ is a prime ideal. I would appreciate any reference to a source where any of the following questions were discussed:

i) Which quotients $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$ do occur as rings of algebraic integers of number fields? ($\mathfrak{p}$ cannot be maximal; what other restrictions are there for $r$ and $\mathfrak{p}$?)

ii) What can be said about the minimal $r$ such that a given $\mathcal{O}_K$ is isomorphic to $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$? (It is at most $[K : \mathbb{Q}]$ and is sometimes larger than $1$; given $d$, do all values $1, 2, ..., d$ occur for some $K$ of degree $[K : \mathbb{Q}] = d$?)

iii) What can be said about the minimal number of generators of $\mathfrak{p}$, when $\mathcal{O}_K \cong \mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$ and $r$ is minimal? (Does it sometimes have to be "very large"?)

Thank you!