$(1) \iff (2) \implies (3) \iff (4) \iff (5)$ holds. If $R/\m$ is infinite, then $(5) \iff (7)$ and $(6) \implies (4)$ hold. If $R$ has an infinite coefficient field $k\subseteq R$, then all the conditions are equivalent.

Here we use the notion of almost Gorenstein rings in the sense of [@GMP]. Also we denote by $e(S)$ the multiplicity of $S$ and by $\edim S$ the embedding dimension of $S$. A Gorenstein ring of dimension one satisfying $e(S)= \edim S+1$ are called *a ring of almost minimal multiplicity* or a *Gorenstein ring of minimal multiplicity*, and studied by J. D. Sally [@Sally]. The invariant $\rho(R)$ is the *canonical index* of $R$, introduced by Ghezzi-Goto-Hong-Vasconcelos [@GGHV].

The second main theorem of this paper is the following, which is a one-dimensional analogue of Theorem \[Anan\].

\[thB\] Let $(R,\m)$ be a complete one-dimensional Cohen-Macaulay local ring. Consider the following conditions.

-   $bg(R)\leq 2$.

-   There exists a self-dual ideal $\mathfrak{a}\subseteq R$ such that $\ell_R(R/\mathfrak{a})\leq 2$.

Then $(1)$ implies $(2)$. The implication $(2) \implies (1)$ also holds if $R$ has an infinite coefficient field $k\subseteq R$.

In the view of Theorem \[thA\], local rings with self-dual maximal ideal are naturally constructed from Gorenstein local 