the analytically unramified case.

Let $(R,\m)$ be a complete one-dimensional local ring. Assume $R$ contains an infinite coefficient field. If there exists a module-finite birational extension $R\subseteq T$ with a Gorenstein ring $T$, Then $bg(R)\leq l(R/aT)$ for any non-zero divisor $a\in T:R$ of $T$. In particular, if $R$ is analytically unramified, then $bg(R)\leq l(R/R:\widetilde R)<\infty$.

Since $T$ is Gorenstein, the $R$-module $aT\cong T$ is self-dual. So we can apply Lemma \[lem23\] for $I=aT$. If $R$ is analytically unramified, then the normalization $\widetilde R$ of $R$ in $Q(R)$ is Gorenstein, and $R\subseteq \widetilde R$ is finite birational. Take any non-zero element $a\in R:\widetilde R$. Then $bg(R)\leq l(R/a(R:\widetilde R))$. Since $R:\widetilde R$ is contained in $R$, we have an inequality $(R/a(R:\widetilde R))\le l(R/R:\widetilde R)$.

Ananthnarayan [@Ananthnarayan] shows the following inequalities hold for an artinian local ring $R$.

$$\ell_R(R/\omega^*(\omega))\leq \min\{\ell_R(R/I)\mid I\cong I^\dag\} \leq g(R).$$

Here $\omega^*(\omega)$ is the trace ideal of $\omega$; see Definition \[deftr\].

As analogies of these inequalities, the followings are natural questions.

\[Q1\] Let $(R,\m)$ be a one-dimensional Cohen-Macaulay local ring. Is an equality $$bg(R)\geq\inf\{\ell_R(R/I)\mid I\cong I^\dag\}$$ hold true?

\[Q2\] Let $(R,\m)$ be a one-dimensional generically Gorenstein local ring. Is an inequality $\ell_R(R/\omega^*(\omega))\leq bg(R)$ hold true?

By our main theorems 