\m_B$. This says that $\m_B$ is an ideal of $R$. Since $bg(B)=1$, $\m_B$ is a self-dual ideal of $B$ by Theorem \[thA\]. Thus, it is also self-dual as $R$-module. One can also have equalities $$\ell_R(R/\m_B)=\ell_B(R/B)+\ell_B(B/\m_B)=2.$$

Let $(R,\m)$ be a one-dimensional local ring. Assume $R$ is complete, equicharacteristic and $bg(R)=n<\infty$. If there exists a Cohen-Macaulay local ring $(B,\m_B)$ with $bg(B)=1$ and module-finite residually rational birational extensions $B\subseteq R\subseteq \m_B:\m_B$ such that $\ell_B(R/B)+1\leq n$. Then, by the same argument of proof of Theorem \[thB\], it follows that $\m_B$ is a self-dual ideal of $R$ satisfying $\ell_R(R/\m_B)\leq n$. In this case, Question \[Q1\] is affirmative for $R$.

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Let $R=k[[t^3,t^{10},t^{11}]]$ be a numerical semigroup ring, where $k$ is a field. We give some observation on Question \[Q1\] fo $R$. $R$ has minimal multiplicity and is not almost Gorenstein. By Corollary, $bg(R)\geq 2$. In the case $bg(R)=2$, Question \[Q1\] is affirmative by Theorem \[thB\]. However, we don’t know whether $bg(R)=2$ or not.

Let $S=k[[t^3,t^{10}]]$ and $\mathfrak{n}$ be its maximal ideal. Then $S$ is Gorenstein and $S\subseteq R$ is module-finite residually rational birational extension with $\ell_S(R/S)=3$. In particular, $bg(R)\leq 3$. Put $B=\mathfrak{n}:\mathfrak{n}=k[[t^3,t^{10},t^{17}]]$. Then $\m_B:\m_B$ is equal to $k[[t^3,t^7]]$ and does not contain $R$. Therefore, we 