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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.
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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

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