module-finite, residually rational and birational. Since $I:I=R$ and $\ell_R(\m/I)\le 1$, we can see that $S$ is Gorenstein and $\End_S(I)\cong R$ by the similar argument in the proof of \[thA\] (3) $\implies$ (1). Furthermore, one has an equality $tI=I^2$, which particularly show that $S$ has minimal multiplicity, that is, $e(S)=\edim S$.

Now consider the case that $bg(R)=1$. Repeating the proof of Theorem \[thA\] (3) $\implies$ (1), there is a canonical ideal $I$ such that if we let $S=k+I$, then $S$ is Gorenstein local and $R\cong \End_S(\mathfrak{n})$, where $\mathfrak{n}$ is the maximal ideal of $S$. Since $R$ is Almost Gorenstein, it was shown in [@GMP Theorem 3.16] that there is a minimal reduction $Q=(t)\subseteq I$ of $I$ in $R$ such that $\ell_R(I^2/QI)\le 1$ and $QI^2=I^3$. Then it follows that $\ell_S(I^2/QI)\le 1$. Using [@Sally2 Proposition 3.3], the equality $e(S)=\edim S+1$ holds.

We give here an example of a ring $R$ with $bg(R)=1$.

Let $R=k[[t^3,t^4,t^5]]$ and $S=k[[t^3,t^4]]$ be numerical semigroup rings, where $k$ is a field. Then the natural inclusion $S\subseteq R$ is a module-finite birational extension of local rings with the same coefficient field. The colength $\ell_S(R/S)$ is equal to $1$. Since $R$ is non-Gorenstein and $S$ is Gorenstein, we have $bg(R)=1$.

We 