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On the Integral Degree of Integral Ring Extensions

Part of: Field extensions Ring extensions and related topics Integral domains

Published online by Cambridge University Press:  20 August 2018

José M. Giral ,
Liam O'Carroll ,
Francesc Planas-Vilanova  and
Bernat Plans
José M. Giral*
Affiliation:
Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain (giral@ub.edu)
Liam O'Carroll
Affiliation:
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, Edinburgh EH9 3DF, UK
Francesc Planas-Vilanova
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, ETSEIB, E-08028 Barcelona, Spain (francesc.planas@upc.edu; bernat.plans@upc.edu)
Bernat Plans
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, ETSEIB, E-08028 Barcelona, Spain (francesc.planas@upc.edu; bernat.plans@upc.edu)
*
*Corresponding author.
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Abstract

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

Keywords

integral extensionintegral degreeintegral closurefield extensionDedekind domainNagata ring

MSC classification

Primary: 13B21: Integral dependence; going up, going down
Secondary: 13B22: Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.) 13G05: Integral domains 12F05: Algebraic extensions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 25 - 46
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Bourbaki, N., Commutative Algebra. Chapters 1–7. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin) (Springer-Verlag, Berlin, 1998).Google Scholar
2.Bourbaki, N., Algebra. Chapters 4–7. Translated from the 1981 French edition by P. M. Cohn and J. Howie. Reprint of the 1990 English edition (Springer, Berlin). Elements of Mathematics (Berlin) (Springer-Verlag, Berlin, 2003).Google Scholar
3.Giral, J. M. and Planas-Vilanova, F., Integral degree of a ring, reduction numbers and uniform Artin–Rees numbers, J. Algebra 319 (2008), 3398–3418.Google Scholar
4.Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, Volume 336 (Cambridge University Press, Cambridge, 2006).Google Scholar
5.Jacobson, N., Structure theory for algebraic algebras of bounded degree, Ann. of Math. 46 (1945), 695–707.Google Scholar
6.Jacobson, N., Lectures in abstract algebra. III. Theory of fields and Galois theory. Second corrected printing, Graduate Texts in Mathematics, Volume 32 (Springer-Verlag, New York-Heidelberg, 1975).Google Scholar
7.Janusz, G. J., Algebraic number fields, 2nd edn, Graduate Studies in Mathematics, Volume 7 (American Mathematical Society, 1996).Google Scholar
8.Jensen, C. U., Ledet, A. and Yui, N., Generic polynomials. Constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications, Volume 45 (Cambridge University Press, Cambridge, 2002).Google Scholar
9.Kaplansky, I., On a problem of Kurosch and Jacobson, Bull. Amer. Math. Soc. 52 (1946), 496–500.Google Scholar
10.Kunz, E., Introduction to commutative algebra and algebraic geometry, Translated from the German by Michael Ackerman. With a preface by David Mumford (Birkhäuser, Boston, MA, 1985).Google Scholar
11.Kurosch, A., Ringtheoretische Probleme, die mit dem Burnsideschen Problem über periodische Gruppen in Zusammenhang stehen, Bull. Acad. Sci. URSS. Sr. Math. [Izvestia Akad. Nauk SSSR] 5 (1941), 233–240.Google Scholar
12.Levitzki, J., On a problem of A. Kurosch, Bull. Amer. Math. Soc. 52 (1946), 1033–1035.Google Scholar
13.MacKenzie, R. and Scheuneman, John, A number field without a relative integral basis, Amer. Math. Monthly 78 (1971), 882–883.Google Scholar
14.Matsumura, H., Commutative algebra, 2nd edn, Mathematics Lecture Note Series, Volume 56 (Benjamin Cummings Publishing, Reading, MA, 1980).Google Scholar
15.Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, Volume 8 (Cambridge University Press, Cambridge, 1986).Google Scholar
16.Narkiewicz, W., Elementary and analytic theory of algebraic numbers, 3rd edn, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2004).Google Scholar
17.Sally, J. and Vasconcelos, W. V., Stable rings, J. Pure Appl. Algebra 4 (1974), 319–336.Google Scholar
18.Sharp, R. Y. and Vámos, P., Baire's category theorem and prime avoidance in complete local rings, Arch. Math. (Basel) 44(3) (1985), 243–248.Google Scholar
19.Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. Algorithms and Computation in Mathematics, Volume 2 (Springer-Verlag, Berlin, 1998).Google Scholar
20.Voight, J., Rings of low rank with a standard involution, Illinois J. Math. 55 (2011), 1135–1154.Google Scholar