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

## TL;DR

This paper develops explicit methods for generating ideals in certain subalgebras of polynomial rings over a field, including classes larger than numerical semigroup rings, with results on minimal generators of maximal ideals.

## Contribution

It introduces efficient generation techniques for ideals in core subalgebras of polynomial rings, expanding beyond numerical semigroup rings and analyzing minimal generators.

## Key findings

- Minimal number of generators of maximal ideals is 1, 2, or μ(H).
- Results generalize classical theorems of Forster and Swan.
- Includes explicit generation methods for a broad class of subalgebras.

## Abstract

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.