Efficient generation of ideals in core subalgebras of the polynomial ring $k[t]$ over a field $k$
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- by Naoki Endo, Shiro Goto, Naoyuki Matsuoka and Yuki Yamamoto PDF
- Proc. Amer. Math. Soc. 148 (2020), 3283-3292 Request permission
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.References
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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
- MR Author ID: 1086482
- ORCID: 0000-0001-9343-7161
- 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
- MR Author ID: 192104
- 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
- MR Author ID: 803509
- 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
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3283-3292
- MSC (2010): Primary 13A15, 13B25, 13B22
- DOI: https://doi.org/10.1090/proc/15032
- MathSciNet review: 4108838