By Theorem \[thA\], it holds that either $R$ is Gorenstein or $\m\cong \m^\dag$. In the case that $R$ is Gorenstein, it holds that $e(R)\le 2$ and so $\m$ is self-dual by [@Ooishi Theorem 2.6].

\(5) $\implies$ (7): Assume $R$ is almost Gorenstein and has minimal multiplicity. Then we already saw that $m$ is self-dual. It follows from [@GMP Theorem 3.16] that $\rho(R)\le 2$.

\(7) $\implies$ (5): It is enough to show that $\rho(R)\le 2$. Recall that $\rho(R)$ is the reduction number of a canonical ideal of $R$ ([@GGHV Definition 4.2]). So if $\rho(R)\le 1$, then $R$ is Gorenstein ([@GMP Theorem 3.7]). Assume $\rho(R)=2$. Combining [@DR Theorem 3.7, Proposition 3.8] and Theorem \[thA\], we obtain that $R$ is almost Gorenstein and has minimal multiplicity.

Finally, we deal with the assumption that $R$ contains a infinite field $k$ isomorphic to $R/\m$ via $R\to R/\m$.

\(4) $\implies$ (1): This follows directly from Theorem \[thA\].

\(1) $\implies$ (6): First we consider the case that $R$ is Gorenstein (i.e. $bg(R)=0$). In this case, $e(R)\le 2$ and $\edim R\le 2$ by the assumption. Take a minimal reduction $Rt$ of $\m$. Then $\m^2=t\m$. In particular, $\ell_R(\m/I)=\ell_R(\m/I+\m^2)\le 1$. Put $I=Rt$ and $S=k+I$. Then the ring-extension $S\subseteq R$ is 