Quasi-factorially closed subalgebras of Laurent polynomial rings

TL;DR

This paper characterizes quasi-factorially closed subalgebras of Laurent polynomial rings using lattice and monoid structures, providing criteria for when these subalgebras are qfc, and relates these notions to other algebraic properties.

Contribution

It offers a complete lattice-theoretic characterization of qfc subalgebras in Laurent polynomial rings and introduces an invariant to identify qfc subalgebras.

Findings

R[M] is qfc iff the group generated by M is a direct summand of Z^n.

When n=1, M is a numerical semigroup iff R[M] is qfc.

Finite Gap(A) implies A is qfc in B.

Abstract

Let $R$ be a domain and $B=R[x_1^{\pm1},\ldots,x_n^{\pm1}]$ the Laurent polynomial ring over $R$. In this paper we study pre-factorially closed (pfc) and quasi-factorially closed (qfc) $R$-subalgebras of $B$, 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 $A=R[M]$ associated with submonoids $M\subset \mathbb{Z}^n$. We prove that $R[M]$ is qfc in $B$ if and only if the group generated by $M$ is a direct summand of $\mathbb{Z}^n$. This provides a complete characterization of the qfc property in terms of the lattice structure of the associated group. As a consequence, when $n=1$ and $M\subset\mathbb{N}$, the algebra $R[M]$ is qfc in $B$ precisely when $M$ is a numerical semigroup. For a general $R$-subalgebra $A\subset B$, we introduce an invariant $\mathrm{Gap}(A)$. We show that if $\mathrm{Gap}(A)$ is finite, then $A$ is qfc in $B$. 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 $B$, and normality.