Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-14T14:15:40.503Z Has data issue: false hasContentIssue false
  • English
  • Français

Nearly integral homomorphisms of commutative rings

Published online by Cambridge University Press:  17 April 2009

David E. Dobbs
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN. 37996–1300United States of America.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion R → T is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Barucci, V., Dobbs, D.E. and Mulay, S.B., ‘Integrally closed factor domains’, Bull. Austral. Math. Soc. 37 (1988), 353–366.CrossRefGoogle Scholar
[2]Bourbaki, N., Commutative Algebra (Addison-Wesley, Reading, Mass., 1972).Google Scholar
[3]Demazure, M. and Gabriel, P., Introduction to Algebraic Geometry and Algebraic Groups (North Holland, Amsterdam, 1980).Google Scholar
[4]Dobbs, D.E., ‘Coherence, ascent of going-down and pseudo-valuation domains’, Houston J. Math. 4 (1978), 551–567.Google Scholar
[5]Dobbs, D.E., ‘Lying-over pairs of commutative rings’, Canad. J. Math. 33 (1981), 454–475.CrossRefGoogle Scholar
[6]Fontana, M., ‘Topologically defined classes of commutative rings’, Ann. Mat. Pura Appl. 123 (1980), 331–355.CrossRefGoogle Scholar
[7]Gilmer, R. and Hoffmann, J.F., ‘A characterization of Prüfer domains in terms of polynomials’, Pacific J. Math. 60 (1975), 81–85.CrossRefGoogle Scholar
[8]Greenberg, B., ‘Coherence in Cartesian squares’, J. Algebra 50 (1978), 12–25.CrossRefGoogle Scholar
[9]Grothendieck, A. and Dieudonné, J., Eléments de géométrie algébrique, I (Springer-Verlag, Berlin, Heidelberg, New York, 1971).Google Scholar
[10]Kaplansky, I., Commutative rings, rev. ed. (Univ. of Chicago Press, 1974).Google Scholar
[11]Richman, F., ‘Generalized quotient rings’, Proc. Amer. Math. Soc. 18 (1965), 794–799.CrossRefGoogle Scholar