REFLECTION OF SOME QUASI-LOCAL DOMAINS

Abstract

If (R, M) and (S, N) are quasi-local domains and f : R → S is a ring homomorphism, then f is said to be a strong local homomorphism if f(M) = N. Let PVD be the category whose objects are the pseudo-valuation domains that are not fields and whose morphisms are the strong local homomorphisms; let VD be the full subcategory of PVD whose objects are all the valuation domains that are not fields. Then VD is shown to be a reflective subcategory of PVD. This result is extended by obtaining a reflective conclusion for a category whose class of objects properly contains all pseudo-valuation domains.