# Localizing Global Properties to Individual Maximal Ideals

Chapter

## 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:

Primary: 13A15, 13G05

## References

1. 1.
D.D. Anderson, M. Zafrullah, Independent locally-finite intersections of localizations. Houston J. Math. 25, 433–452 (1999)
2. 2.
M. Fontana, E. Houston, and T. Lucas, Toward a classification of prime ideals in Prüfer domains. Forum Math. 22, 741–766 (2010)
3. 3.
M. Fontana, E. Houston, T. Lucas, Factoring Ideals in Integral Domains, Lecture Notes of the Unione Matematica Italiana, vol. 14 (Springer, Berlin, 2013)
4. 4.
R. Gilmer, Overrings of Prüfer domains, J. Algebra 4, 331–340 (1966)
5. 5.
R. Gilmer Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, vol. 90 (Queen’s University Press, Kingston, 1992)Google Scholar
6. 6.
R. Gilmer, W. Heinzer, Overrings of Prüfer domains. II. J. Algebra 7, 281–302 (1967)
7. 7.
E. Matlis, Cotorsion Modules, Mem. American Mathematical Society No, vol. 49, (Providence RI, 1964)Google Scholar
8. 8.
B. Olberding, Globalizing local properties of Prüfer domains. J. Algebra 205, 480–504 (1998)