Rings of Krull dimension one I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate a lot to know if there is some result of the kind:
$$ \dim(A)=1 \iff ~?$$
Thanks in advance.

An integral extension $S$ of a ring $R$ of Krull dimension $1$ has Krull dimension $1$. This is because any 3-chain of prime ideals $P_1 \subsetneq P_2 \subsetneq P_3$ induces an inclusion $p_1 \subset p_2 \subset p_3$ (where we define $p_j := R \cap P_j$ for $j = 1, 2, 3$), of which not all inclusions can be proper because of what we assumed about $R$, and then we only have to use that between two prime ideals lying over the same prime ideal there are no proper inclusions.
This, of course, generalises to show that the Krull dimension is constant when passing to an integral extension.
