Mathematische Nachrichten· 2026Q1
Quasi‐Factorially Closed Subalgebras of Laurent Polynomial Rings
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- 2026year
Short summary
A new localization criterion is established for quasi-factorially closed (qfc) subalgebras of Laurent polynomial rings, enabling a complete characterization of the qfc property for monoid algebras.
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Key points
- A localization criterion for the quasi-factorially closed (qfc) property of R-subalgebras in Laurent polynomial rings R[x, x^-1] is established.
- For monoid algebras R[M] associated with submonoids M of G, R[M] is qfc in R[G] iff the group generated by M is a direct summand of G.
- When R is a domain and M is a nonzero submonoid of Z, R[M] is qfc in R[Z] precisely when M is a numerical semigroup.
- An invariant is introduced for general R-subalgebras, showing it's qfc if finite.
AI-generated from the title and abstract; the full text is not read.
Abstract
ABSTRACT Let be a domain and the Laurent polynomial ring over . In this paper we study pre‐factorially closed (pfc) and quasi‐factorially closed (qfc) ‐subalgebras of , which generalize the notion of factorially closed subalgebras. We first establish a localization criterion for the qfc property. Using this criterion, we investigate monoid algebras associated with submonoids . We prove that is qfc in if and only if the group generated by is a direct summand of . This provides a complete characterization of the qfc property in terms of the lattice structure of the associated group. As a consequence, when and is a nonzero submonoid of , the algebra is qfc in precisely when is a numerical semigroup. For a general ‐subalgebra , we introduce an invariant . We show that if is finite, then is qfc in . Moreover, we clarify how the pfc and qfc conditions are related to other notions that naturally appear for subalgebras, such as retracts, being algebraically closed in , and normality.
The authors' abstract, as published at the source. Mathematische Nachrichten, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics