$\m=t(K_R:\m)$. Put $I=tK_R$. Then $I=t(K_R:R)\subseteq t(K_R:\m)=\m$ and thus $I$ is an ideal of $R$. Set $S=k+I$. The extension $S\subseteq R$ is module-finite, residually rational and birational. We claim that $S$ is Gorenstein, and prove this in the followings. $I=tK_R$ implies $I:I=tK_R:tK_R=K_R:K_R=R$. Thus $R$ is the endomorphism ring of the maximal ideal $I$ of $S$. In particular, the colength $\ell_S(S/R)$ is equal to the Cohen-Macaulay type of $S$. Since $R$ and $S$ have same residue field $k$, we can see the equalities $\ell_S(R/S)=\ell_S(\m/I)=\ell_R(\m/I)$. On the other hand, we have $$\ell_R(\m/I)=\ell_R(t(K_R:\m)/t(K_R:R))=\ell_R((K_R:\m)/(K_R:R))=\ell_R(R/\m)=1.$$ It follows that $S$ has Cohen-Macaulay type $1$, that is, $S$ is Gorenstein.

We put the following lemma here, which will be used in the proof of Corollary \[co1\].

\[lem16\] Let $(R,\m)$ be a Cohen-Macaulay generically Gorenstien local ring of dimension one having a canonical module. Assume $R$ is not a ，. Then

-   $R$ has minimal multiplicity if and only if $\m\cong \m:\m$.

-   $R$ is almost Gorenstein in the sense of [@GMP] if and only if $\m^\dag\cong\m:\m$.

See [@Ooishi Proposition 2.5] and [@Ko Theorem 2.14] respectively.

We give a proof of Corollary \[co1\] as follows.

The implications (3) $\iff$ (4) $\iff$ (5) follow immediately from Lemma \[lem16\].

\(1) $\iff$ 