Quasi-factorially closed subalgebras of Laurent polynomial rings
Shinya Kumashiro, Takanori Nagamine
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.