$1$-dimensional local Cohen--Macaulay ring whose normalisation is a module finite extension

Question

Let $(R,\mathfrak m)$ be a local $1$-dimendlsional Cohen-Macaulay ring with total ring of fractions (https://en.m.wikipedia.org/wiki/Total_ring_of_fractions) $Q(R)$. Let $\bar R$ be the integral closure of $R$ inside $Q(R)$ . If $\bar R$ is a module finite extension of $R$, then is $R$ necessarily reduced?

Answer 1

I assume that $R$ is Noetherian. If $R$ is not reduced, let $z\neq 0$ be a nilpotent element and let $x$ a nonzero divisor in the maximal ideal. Then $z/x^n\in \overline{R}$ for all $n$. Take the suubmodule generated by all these elements. It is easy to see that this is not finitely generated.