In Lang's "Algebra" (chap 16, p 614) he states the following without proof:
"Let $R$ be some ring in a finite extension $K$ of $\mathbb{Q}$, and such that $R$ is a finite module over $\mathbb{Z}$ but not integrally closed. Let $R'$ be its integral closure. Let $\mathfrak{p}$ be a maximal ideal of $R$ and suppose that $\mathfrak{p}R'$ is contained in two distinct maximal ideals $\mathfrak{P}_1$ and $\mathfrak{P}_2.$ Then it can be shown that $R'$ is not flat over $R,$ otherwise $R'$ would be free over the local ring $R_{\mathfrak{p}},$ and the rank would have to be 1, thus precluding the possibility of the two primes $\mathfrak{P}_1$ and $\mathfrak{P}_2.$"
I would appreciate either a proof of Lang's assertion, or to be guided to a relevant source.
It is shown later in the chapter that a flat, finitely generated module over a local ring is free, but I am not able to use this to prove the above statement and I suspect that other arguments are needed.
The situation described by Lang occurs naturally in algebraic number theory where we take $R= \mathbb{Z}$ and $R'= \mathcal{O}_K$ to the the ring of integers of the number field $K,$ so I am surprised I haven't been able to find anything on the internet so far.
I don't understand Lang's assertion "$R'$ would be free over the local ring $R_{\mathfrak p}$". In fact, I can't understand the whole passage.
In fact, $R'$ is never flat (unless $R=R'$). Suppose $R\subset R'$ is flat. Then it is faithfully flat and since $R$ and $R'$ have the same field of fractions we get $R=R'$. (See Matsumura, CRT, Exercise 7.2.)