Localizing Global Properties to Individual Maximal Ideals

Abstract

We consider three related questions. Q 1: Given a global property G of a domain R, what does a particular maximal ideal M of R “know” about the property with regard to the ideals I ⊆ M and elements t ∈ M? Suppose P is such a property corresponding to G. Q 2: If each maximal ideal knows it has property P, does R have the corresponding global property G? Q 3: If at least one maximal ideal knows it has property P, does R have the global property G? We assume that if I ⊆ M, then M can tell when a particular element t ∈ M is contained in I and when it isn’t. Thus for a pair of ideals I and J contained in M, M knows when \(I \subseteq J\). In addition, this allows M to understand the intersection of ideals it contains. In some cases, if a single maximal ideal knows P, then R will satisfy G. For example, there are such Ps for G ∈ {PIDs, Noetherian domains, Domains with ACCP, Domains with finite character}.