Let $R$ be a Noetherian domain of Krull dimension $\leq 1$ with the quotient field $K$. Let $T$ be an intermediate ring between $R$ and $K$. Then, why is $T$ of Krull dimension $\leq 1$?
(Reference: Kaplansky, Commutative Rings, p.61.)
The proof given in the book shows that $T$ is a Noetherian domain and $T/I$ is a Noetherian $R$-module for each non-zero ideal $I$ of $T$.
However, I do not get how to conclude $\dim(T)\leq 1$ from the facts given above. How?
This would be true if we can show that $T/I$ is an Artinian ring for each nonzero ideal $I$, but I do not know how this can be proved.