Quasi‐Factorially Closed Subalgebras of Laurent Polynomial Rings

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.