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Pairs of Rings with the Same Prime Ideals

Published online by Cambridge University Press:  20 November 2018

David F. Anderson  and
David E. Dobbs
David F. Anderson
Affiliation:
University of Tennessee, Knoxville, Tennessee
David E. Dobbs
Affiliation:
University of Tennessee, Knoxville, Tennessee
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There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 362 - 384
Copyright
Copyright © Canadian Mathematical Society 1980

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