Chapter

Commutative Algebra

pp 239-254

Date:

Localizing Global Properties to Individual Maximal Ideals

  • Thomas G. LucasAffiliated withDepartment of Mathematics and Statistics, University of North Carolina Charlotte Email author 

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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 IM and elements tM? 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 IM, then M can tell when a particular element tM 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}.

Keywords

Integral domain Maximal ideal

MSC(2010) classification:

[2010]Primary: 13A15, 13G05