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Efficient generation of ideals in core subalgebras of the polynomial ring $k[t]$ over a field $k$
- Source :
- Proceedings of the American Mathematical Society. 148:3283-3292
- Publication Year :
- 2020
- Publisher :
- American Mathematical Society (AMS), 2020.
-
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.<br />Comment: 10 pages
- Subjects :
- Physics
Mathematics::Operator Algebras
Semigroup
Applied Mathematics
General Mathematics
Polynomial ring
Field (mathematics)
Mathematics - Commutative Algebra
Commutative Algebra (math.AC)
13A15, 13B25, 13B22
Combinatorics
Integer
Numerical semigroup
Core (graph theory)
FOS: Mathematics
Integral closure of an ideal
Subjects
Details
- ISSN :
- 10886826 and 00029939
- Volume :
- 148
- Database :
- OpenAIRE
- Journal :
- Proceedings of the American Mathematical Society
- Accession number :
- edsair.doi.dedup.....643bc8b8559e41f5a6bf77413bbb286b