Abstract
In the first section, we introduce the notion of an h-local maximal ideal as a maximal ideal M of a domain R such that \(\Theta (M){R}_{M} = K\) (the quotient field of R). The second section deals with independent pairs of overrings of a domain R. In the case R can be realized as the intersection of a pair of independent overrings, we show that R shares various factorization properties with these overrings. For example, R has weak factorization if and only if both overrings have weak factorization. The third section introduces Jaffard families and Matlis partitions. Just as domains of Dedekind type are the same as h-local domains, a domain R can be realized as an intersection of the domains of a Jaffard family if and only if its set of maximal ideals can be partitioned into a Matlis partition (definitions below). As in the second section, if \(R ={ \bigcap \nolimits }_{\alpha \in \mathcal{A}}{S}_{\alpha }\) where \(\{{S{}_{\alpha }\}}_{\alpha \in \mathcal{A}}\) is a Jaffard family, then R satisfies a particular factoring property if and only if each S α satisfies the same factoring property. The last section is devoted to constructing examples using various Jaffard families.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
D.D. Anderson, Non-finitely generated ideals in valuation domains. Tamkang J. Math. 18, 49–52 (1987)
D.D. Anderson, J. Huckaba, I. Papick, A note on stable domains. Houst. J. Math. 13, 13–17 (1987)
D.D. Anderson, L. Mahaney, Commutative rings in which every ideal is a product of primary ideals. J. Algebra 106, 528–535 (1987)
D.D. Anderson, M. Zafrullah, Independent locally-finite intersections of localizations. Houst. J. Math. 25, 433–452 (1999)
D.F. Anderson, D.E. Dobbs, M. Fontana, On treed Nagata rings. J. Pure Appl. Algebra 61, 107–122 (1989)
E. Bastida, R. Gilmer, Overrings and divisorial ideals of rings of the form D + M. Mich. Math. J. 20, 79–95 (1973)
S. Bazzoni, Class semigroups of Prüfer domains. J. Algebra 184, 613–631 (1996)
S. Bazzoni, Clifford regular domains. J. Algebra 238, 703–722 (2001)
S. Bazzoni, L. Salce, Warfield domains. J. Algebra 185, 836–868 (1996)
N. Bourbaki, Algèbre Commutative, Chapitres I et II (Hermann, Paris, 1961)
J. Brewer, W. Heinzer, On decomposing ideals into products of comaximal ideals. Comm. Algebra 30, 5999–6010 (2002)
H.S. Butts, Unique factorization of ideals into nonfactorable ideals. Proc. Am. Math. Soc. 15, 21 (1964)
H.S. Butts, R. Gilmer, Primary ideals and prime power ideals. Can. J. Math. 18, 1183–1195 (1966)
H.S. Butts, R.W. Yeagy, Finite bases for integral closures. J. Reine Angew. Math. 282, 114–125 (1976)
P.-J. Cahen, T. Lucas, The special trace property, in Commutative Ring Theory (Fès, Morocco, 1995), Lecture Notes in Pure and Applied Mathematics, vol. 185 (Marcel Dekker, New York, 1997), pp. 161–172
D.E. Dobbs, M. Fontana, Locally pseudo-valuation domains. Ann. Mat. Pura Appl. 134, 147–168 (1983)
P. Eakin, A. Sathaye, Prestable ideals. J. Algebra 41, 439–454 (1976)
S. El Baghdadi, S. Gabelli, Ring-theoretic properties of PvMDs. Comm. Algebra 35, 1607–1625 (2007)
M. Fontana, E. Houston, T. Lucas, Factoring ideals in Prüfer domains. J. Pure Appl. Algebra 211, 1–13 (2007)
M. Fontana, E. Houston, T. Lucas, Toward a classification of prime ideals in Prüfer domains. Forum Math. 22, 741–766 (2010)
M. Fontana, J. Huckaba, I. Papick, Divisorial prime ideals in Prüfer domains. Can. Math. Bull. 27, 324–328 (1984)
M. Fontana, J. Huckaba, I. Papick, Some properties of divisorial prime ideals in Prüfer domains. J. Pure Appl. Algebra 39, 95–103 (1986)
M. Fontana, J. Huckaba, I. Papick, Domains satisfying the trace property. J. Algebra 107, 169–182 (1987)
M. Fontana, J. Huckaba, I. Papick, Prüfer Domains (Marcel Dekker, New York, 1997)
M. Fontana, J. Huckaba, I. Papick, M. Roitman, Prüfer domains and endomorphism rings of their ideals. J. Algebra 157, 489–516 (1993)
M. Fontana, N. Popescu, Sur une classe d’anneaux qui généralisent les anneaux de Dedekind. J. Algebra 173, 44–66 (1995)
L. Fuchs, L. Salce, Modules Over Non-Noetherian Domains, Math. Surveys & Monographs, vol. 84 (American Mathematical Society, Providence, 2001)
S. Gabelli, A class of Prüfer domains with nice divisorial ideals, in Commutative Ring Theory (Fès, Morocco, 1995), Lecture Notes in Pure and Applied Mathematics, vol. 185 (Marcel Dekker, New York, 1997), pp. 313–318
S. Gabelli, Generalized Dedekind domains, in Multiplicative Ideal Theory in Commutative Algebra (Springer, New York, 2006), pp. 189–206
S. Gabelli, E. Houston, Coherent-like conditions in pullbacks. Mich. Math. J. 44, 99–123 (1997)
S. Gabelli, N. Popescu, Invertible and divisorial ideals of generalized Dedekind domains. J. Pure Appl. Algebra 135, 237–251 (1999)
R. Gilmer, Integral domains which are almost Dedekind. Proc. Am. Math. Soc. 15, 813–818 (1964)
R. Gilmer, Overrings of Prüfer domains. J. Algebra 4, 331–340 (1966)
R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, vol. 90 (Queen’s University Press, Kingston, 1992)
R. Gilmer, W. Heinzer, Overrings of Prüfer domains. II. J. Algebra 7, 281–302 (1967)
R. Gilmer, W. Heinzer, Irredundant intersections of valuation domains. Math. Z. 103, 306–317 (1968)
R. Gilmer, J. Hoffmann, A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60, 81–85 (1975)
R. Gilmer, J. Huckaba, The transform formula for ideals. J. Algebra 21, 191–215 (1972)
H. Gunji, D.L. McQuillan, On rings with a certain divisibility property. Mich. Math. J. 22, 289–299 (1975)
F. Halter-Koch, Kronecker function rings and generalized integral closures. Comm. Algebra 31, 45–59 (2003)
F. Halter-Koch, Clifford semigroups of ideals in monoids and domains. Forum Math. 21, 1001–1020 (2009)
J.H. Hays, The S-transform and the ideal transform. J. Algebra 57, 223–229 (1979)
J. Hedstrom, E. Houston, Pseudo-valuation domains. Pac. J. Math. 75, 137–147 (1978)
W. Heinzer, Integral domains in which each non-zero ideal is divisorial. Mathematika 15, 164–170 (1968)
W. Heinzer, J. Ohm, Locally Noetherian commutative rings. Trans. Am. Math. Soc. 158, 273–284 (1971)
W. Heinzer, I. Papick, The radical trace property. J. Algebra 112, 110–121 (1988)
M. Henriksen, On the prime ideals of the ring of entire functions. Pac. J. Math. 3, 711–720 (1953)
E. Houston, S.-E. Kabbaj, T. Lucas, A. Mimouni, When is the dual of an ideal a ring? J. Algebra 225, 429–450 (2000)
W.C. Holland, J. Martinez, W.Wm. McGovern, M. Tesemma, Bazzoni’s conjecture. J. Algebra 320, 1764–1768 (2008)
J. Huckaba, in Commutative Rings with Zero Divisors, Pure and Applied Mathematics, vol. 117 (Marcel Dekker, New York, 1988)
J. Huckaba, I. Papick, When the dual of an ideal is a ring? Manuscripta Math. 37, 67–85 (1982)
P. Jaffard, Théorie arithmétique des anneau du type de Dedekind. Bull. Soc. Math. France 80, 61–100 (1952)
C. Jayram, Almost Q-rings. Arch. Math. (Brno) 40, 249–257 (2004)
I. Kaplansky, Commutative Rings (Allyn and Bacon, Boston, 1970)
K. Kubo, Über die Noetherschen fünf Axiome in kommutativen Ringen. J. Sci. Hiroshima Univ. Ser. A 10, 77–84 (1940)
J. Lipman, Stable rings and Arf rings. Am. J. Math. 93, 649–685 (1971)
K.A. Loper, On Prüfer non-D-rings. J. Pure Appl. Algebra 96, 271–278 (1994)
A. Loper, T. Lucas, Factoring ideals in almost Dedekind domains. J. Reine Angew. Math. 565, 61–78 (2003)
T. Lucas, The radical trace property and primary ideals. J. Algebra 184, 1093–1112 (1996)
E. Matlis, Cotorsion Modules, vol. 49 (Memoirs of the American Mathematical Society, Providence RI, 1964)
E. Matlis, Reflexive domains. J. Algebra 8, 1–33 (1968)
K. Matsusita, Über ein bewertungstheoretisches Axiomensystem für die Dedekind-Noether-sche Idealtheorie. Jpn. J. Math. 19, 97–110 (1944)
W.Wm. McGovern, Prüfer domains with Clifford class semigroup. J. Comm. Algebra 3, 551–559 (2011)
S. Mori, Allgemeine Z.P.I.-Ringe. J. Sci. Hiroshima Univ. A 10, 117–136 (1940)
N. Nakano, Idealtheorie in einem speziellen unendlichen algebraischen Zahlkörper. J. Sci. Hiroshima Univ. A 16, 425–439 (1953)
E. Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionen-körpern. Math. Ann. 96, 26–61 (1927)
B. Olberding, Globalizing local properties of Prüfer domains. J. Algebra 205, 480–504 (1998)
B. Olberding, Factorization into prime and invertible ideals. J. Lond. Math. Soc. 62, 336–344 (2000)
B. Olberding, Factorization into radical ideals, in Arithmetical Properties of Commutative Rings and Monoids, Lecture Notes Pure and Applied Mathematics, vol. 241 (Chapman & Hall/CRC, Boca Raton, 2005), pp. 363–377
B. Olberding, Factorization into prime and invertible ideals, II. J. Lond. Math. Soc. 80, 155–170 (2009)
B. Olberding, Characterizations and constructions of h-local domains, in Contributions to Module Theory (Walter de Gruyter, Berlin, 2008), pp. 385–406
E. Popescu, N. Popescu, A characterization of generalized Dedekind domains. Bull. Math. Roumanie 35, 139–141 (1991)
N. Popescu, On a class of Prüfer domains. Rev. Roumaine Math. Pures Appl. 29, 777–786 (1984)
F. Richman, Generalized quotient rings. Proc. Am. Math. Soc. 16, 794–799 (1965)
N.H. Vaughan, R.W. Yeagy, Factoring ideals in semiprime ideals. Can. J. Math. 30, 1313–1318 (1978)
C.A. Wood, On general Z.P.I. rings. Pac. J. Math. 30, 837–846 (1969)
R.W. Yeagy, Semiprime factorizations in unions of Dedekind domains. J. Reine Angew. Math. 310, 182–186 (1979)
M. Zafrullah, t-Invertibility and Bazzoni-like statements. J. Pure Appl. Algebra 214, 654–657 (2010)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fontana, M., Houston, E., Lucas, T. (2012). Factorization and Intersections of Overrings. In: Factoring Ideals in Integral Domains. Lecture Notes of the Unione Matematica Italiana, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31712-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-31712-5_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31711-8
Online ISBN: 978-3-642-31712-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)