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


Authors: Naoki Endo, Shiro Goto, Naoyuki Matsuoka and Yuki Yamamoto
Journal: Proc. Amer. Math. Soc. 148 (2020), 3283-3292
MSC (2010): Primary 13A15, 13B25, 13B22
DOI: https://doi.org/10.1090/proc/15032
Published electronically: May 11, 2020
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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 \mathrm {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.


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Additional Information

Naoki Endo
Affiliation: Global Education Center, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan
Email: naoki.taniguchi@aoni.waseda.jp

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: shirogoto@gmail.com

Naoyuki Matsuoka
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: naomatsu@meiji.ac.jp

Yuki Yamamoto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: yuki.yamamoto5104@gmail.com

DOI: https://doi.org/10.1090/proc/15032
Keywords: Numerical semigroup ring, integral closure of an ideal, rational closed point
Received by editor(s): April 26, 2019
Received by editor(s) in revised form: December 19, 2019
Published electronically: May 11, 2020
Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 17K14176 and Waseda University Grant for Special Research Projects 2019C-444, 2019E-110.
The second author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05112.
The third author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 18K03227.
Communicated by: Claudia Polini
Article copyright: © Copyright 2020 American Mathematical Society