These numerical semigroups are exactly the semigroups obtained by adding one element to a symmetric numerical semigroup. We will show that a numerical semigroup ring has self-dual maximal ideal if and only if the corresponding numerical semigroup is UESY. After that, we also prove that the rings of UESY-numerical semigroup have quasi-decomposable maximal ideal. According to [@NT], an ideal $I$ of $R$ is called [*quasi-decomposable*]{} if there exists a regular sequence $\underline{x}=x_1,\dots,x_t$ such that $I/(\underline{x})$ is decomposable as an $R$-module. Local rings with quasi-decomposable maximal ideal have some interesting properties; we can classify thich subcategories of the singularity category with some assumption on the punctured spectrum ([@NT Theorem 4.5]), and we have results on the vanishings of Ext and Tor ([@NT Section 6]).

In section 5, we characterize the endomorphism ring of a local hypersurface of dimension one, using Theorem \[thA\].

Proof of Theorem \[thA\] and \[thB\]
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In this section, we prove Theorem \[thA\] and \[thB\]. Let $(R,\m)$ be a Noetherian local ring with total quotient ring $Q(R)$. Denote by $\widetilde R$ the integral closure of $R$ in $Q(R)$. A [*fractional ideal*]{} is a finitely generated $R$-submodule $I$ of $Q(R)$ with $IQ(R)=Q(R)$ (i.e. $I$ contains a non zero divisor 