Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-18T00:13:00.062Z Has data issue: false hasContentIssue false
  • English
  • Français

Seminormal rings generated by algebraic integers

Published online by Cambridge University Press:  26 February 2010

David E. Dobbs  and
Marco Fontana
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, U.S.A.
Marco Fontana
Affiliation:
Dipartimento di Matematica, Universita di Roma, “La Sapienza” 00185Roma, Italia.
Get access

Extract

For each algebraic integer α, let ℤα denote the ring of integers of the algebraic number field ℚ(α). There has been continuing interest in finding ring-theoretic conditions characterizing when ℤα coincides with its subring ℤ[α] (cf.[15,18,1,13,12]). One way to extend such work is to consider the intermediate ring ℤ[α]+, the seminormalization (in the sense of [17]) of ℤ[α] in ℤα. Indeed, if we let Iα denote the conductor (ℤ[α]: ℤα), then it is easy to see (cf. Proposition 3.1) that ⅂[α] = ℤα, if, and only if, ℤ[α]+ = ℤα and Iα is a radical ideal of Zα. The condition ℤ[α]+ = ℤα seems worthy of separate attention in view of recent results (cf. [3]) that seminormal rings generated by algebraic integers are “often” automatically of the form ℤα. We show in Proposition 3.3 that the condition ℤ[α]+ = ℤα is equivalent to several universal properties, including notably that the canonical closed surjection Spec (ℤα) → Spec (ℤ[α]) be universally open, be universally going-down, or be a universal homeomorphism.

Type
Research Article
Information
Mathematika , Volume 34 , Issue 2 , December 1987 , pp. 141 - 154
Copyright
Copyright © University College London 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Albu, T.. On a question of Uchida concerning simple finite extensions of Dedekind domains. Osaka J. Math., 16 (1979), 65–69.Google Scholar
2.Andreotti, A. and Bombieri, E.. Sugli omeomorfismi delle varieta algebriche. Ann. Scuola Norm. Sup. Pisa. 23 (1969), 431–450.Google Scholar
3.Angermüller, G.. On the root and integral closure of Noetherian domains of dimension one. J. Algebra, 83 (1983), 437–441.CrossRefGoogle Scholar
4.Dobbs, D. E. and Fontana, M.. Locally pseudo-valuation domains. Ann. Mat. Pura Appl., 134 (1983), 147–168.CrossRefGoogle Scholar
5.Dobbs, D. E. and Fontana, M.. On pseudo-valuation domains and their globalizations, pp. 65–77. Lecture Notes in Pure Appl. Math., vol. 84 (Dekker, New York, 1983).Google Scholar
6.Dobbs, D. E. and Fontana, M.. Universally going-down homomorphisms of commutative rings. J Algebra, 90 (1984), 410–429.CrossRefGoogle Scholar
7.Dobbs, D. E. and Fontana, M.. Universally incomparable ring-homomorphisms. Bull. Austral Math. Soc., 29 (1984), 289–302.CrossRefGoogle Scholar
8.Ferrand, D.. Morphismes entiers universellement ouverts. Manuscript.Google Scholar
9.Gilmer, R.. Multiplicative Ideal Theory (Dekker, New York, 1972).Google Scholar
10.Gilmer, R. and Heitmann, R. C.. On Pic(R[X]) for R seminormal. J Pure Appl. Algebra, 16 (1980), 251–257.CrossRefGoogle Scholar
11.Grothendieck, A. and Dieudonné, J. A.. Eléments de Géométric Algébrique, I (Springer, Berlin, 1971).Google Scholar
12.Hillman, J. A.. Polynomials determining Dedekind domains. Bull. Austral. Math. Soc., 29 (1984), 167–175.CrossRefGoogle Scholar
13.Ishibashi, Y.. On simple integral extensions of normal domains. Bull. Fukuoka Univ. Educ., 29 (1979), 1–8.Google Scholar
14.Kaplansky, I.. Commutative Rings, rev. ed. (Univ. of Chicago Press, 1974).Google Scholar
15.Maury, G.. La condition “intégralement clos” dans quelques structures algebriques. Ann. Sci. Ecole Norm. Sup., 78 (1961), 31–100.CrossRefGoogle Scholar
16.Ooishi, A.. On seminormal rings. Lecture Notes, RIMS, Kyoto Univ., No. 374, (1980), 1–17.Google Scholar
1.Traverso, C.. Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa, 24 (1970), 585–595.Google Scholar
18.Uchida, K.. When is ℤ[α] the ring of the integers? Osaka J. Math., 14 (1977), 155–157.Google Scholar
1.Weiss, E.. Algebraic Number Theory (McGraw-Hill, New York, 1963).Google Scholar