Canonical trace ideal and residue for numerical semigroup rings
In this paper, the authors study numerical semigroup rings. They prove that for a numerical semigroup, \(H\), with three generators and a field, \(K\), the residue of \(K[H]\) is upper bounded by \(g(H)-n(H)\), where \(g(H)\) is the genus of \(H\) and \(n(H)\) is the cardinality of set \(\{x\in H\mid x\leq F(H)\}\) with \(F(H)\) being the Frobenius number of \(H\). Also, for the case of a numerical generators with embedding dimension three, they give a characterization for when the trace of \(H\) is \(H\setminus\{0\}\). Finally, they extend the definition of residue to finitely generated subsemigroups of \(\mathbb{N}\) and give conditions for computing it for shifted numerical semigroups.
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