R$.

The existence of a canonical ideal $I$ of $R$ with $\ell_R(R/I)=2$ is considered by Dibaei-Rahimi [@DR]. Using their notion, the condition (5) above is equivalent to the condition that $\min(S_{\mathfrak{C}_R})\le 2$.

We also remark that Bass’s idea [@B] tells us the importance of the endomorpshism ring $\End_S(\mathfrak{n})$ of the maximal ideal $\mathfrak{n}$ of a Gorenstein local ring $S$ of dimension one. He shew that any torsion-free $S$-module without non-zero free summand can be regarded as a module over $\End_S(\mathfrak{n})$. So we can analyze Cohen-Macaulay representations of $R$ via the ring $\End_S(\mathfrak{n})$ (see also [@LW Chapter 4]).

As a corollary, we can characterize Cohen-Macaulay local rings $R$ of dimension one having minimal multiplicity and satisfying $bg(R)\leq 1$.

\[co1\] Let $(R,\m)$ be a one-dimensional Cohen-Macaulay local ring. Consider the following conditions.

-   $bg(R)\leq 1$ and $R$ has minimal multiplicity.

-   Either $e(R)\leq 2$ or $R$ is almost Gorenstein with $bg(R)=1$.

-   $\m\cong \m^\dag$ and $R$ is almost Gorenstein.

-   $\m\cong \m^\dag$ and $R$ has minimal multiplicity.

-   $R$ is almost Gorenstein and has minimal multiplicity.

-   There exists a Gorenstein local ring $(S,\mathfrak{n})$ of dimension one such that $e(S)\leq \edim S+1$ and $R\cong \End_S(\mathfrak{n})$.

-   $\m\cong \m^\dag$ and $\rho(R)\le 2$.

Then 