Efficient generation of ideals in core subalgebras of the polynomial ring k[t] over a field k

This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call cores of $S$ includes the semigroup rings $k[H]$ of numerical semigroups $H$, but much larger than the class of numerical semigroup rings. For $R=k[H]$ and $M \in \operatorname{Max}R$, our result eventually shows that $\mu_{R}(M) \in \{1,2,\mu(H)\}$ where $\mu_{R}(M)$ (resp. $\mu(H)$) stands for the minimal number of generators of $M$ (resp. $H$), which covers in the specific case the classical result of O. Forster-R. G. Swan.