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Localizing Global Properties to Individual Maximal Ideals
 Thomas G. LucasAffiliated withDepartment of Mathematics and Statistics, University of North Carolina Charlotte Email author
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}.
Keywords
Integral domain Maximal idealMSC(2010) classification:
[2010]Primary: 13A15, 13G05 Title
 Localizing Global Properties to Individual Maximal Ideals
 Book Title
 Commutative Algebra
 Book Subtitle
 Recent Advances in Commutative Rings, IntegerValued Polynomials, and Polynomial Functions
 Pages
 pp 239254
 Copyright
 2014
 DOI
 10.1007/9781493909254_14
 Print ISBN
 9781493909247
 Online ISBN
 9781493909254
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media New York
 Additional Links
 Topics
 Keywords

 Integral domain
 Maximal ideal
 [2010]Primary: 13A15, 13G05
 eBook Packages
 Editors

 Marco Fontana ^{(1)}
 Sophie Frisch ^{(2)}
 Sarah Glaz ^{(3)}
 Editor Affiliations

 1. Dipartimento di Matematica, Università degli Studi Roma Tre
 2. Department of Mathematics, Graz University of Technology
 3. Department of Mathematics, University of Connecticut
 Authors

 Thomas G. Lucas ^{(4)}
 Author Affiliations

 4. Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC, 28223, USA
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