rings, and so their ubiquity is certified. It is interesting to consider how good properties they have, comparing Gorenstein rings. In section 3, we have an observation that a Cohen-Macaulay local ring $(R,\m)$ is nearly Gorenstein in the sense of Herzog-Hibi-Stamate [@HHS] if $\m$ is self-dual. The converse of this is not true in general, however, we have the following result.

\[th313\] Let $(R,\m,k)$ be a Cohen-Macaulay local ring of dimension one. Put $B=\m:\m$. Assume $k$ is infinite.

-   If $B$ is local with Cohen-Macaulay type two and $R$ is nearly Gorenstein, then $R$ is almost Gorenstein and does not satisfy $\m\cong \m^\dag$.

-   If $B$ is local with Cohen-Macaulay type three and $R$ is nearly Gorenstein, then either $R$ is almost Gorenstein or $\m\cong \m^\dag$.

We will provide a proof of Theorem \[th313\] in section 3. One should compare this theorem with the following result of Goto-Matsuoka-Phuong [@GMP Theorem 5.1].

Let $(R,\m,k)$ be a Cohen-Macaulay local ring of dimension one. Put $B=\m:\m$. Then $B$ is Gorenstein if and only if $R$ is almost Gorenstein and has minimal multiplicity.

In section 4, we deal with numerical semigroup rings having self-dual maximal ideal. The definition of UESY-semigroups was given by [@Rosales]. 