Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-28T15:32:19.957Z Has data issue: false hasContentIssue false

Pairs of Rings with the Same Prime Ideals, II

Published online by Cambridge University Press:  20 November 2018

David F. Anderson
Affiliation:
University of Tennessee, Knoxville, Tennessee
David E. Dobbs
Affiliation:
University of Tennessee, Knoxville, Tennessee
Rights & Permissions [Opens in a new window]

Extract

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.

Much of [2] was devoted to studying pairs of subrings AB of a field with the property that A and B have the same prime ideals. In this paper, we continue that investigation, but we no longer assume that A and B are comparable. Interestingly, most of the results of [2] carry over to this more general context. Besides such extensions of [2], additional motivation for the more general context comes from the need to explicate some naturally occurring examples (see Examples 2.5, 3.6, and 4.3).

Section 2 begins by showing that we may reduce to the case in which R is a quasilocal domain with nonzero maximal ideal M and quotient field K. Proposition 2.3 establishes that the set C(R) of all subrings A of K with Spec(A) = Spec(R) forms a complete semilattice.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Alexandra, V. and Popescu, N., On subfields of k(x), Rend. Sem. Mat. Univ. Padova 75 (1986), 257273.Google Scholar
2. Anderson, D. F. and Dobbs, D. E., Pairs of rings with the same prime ideals, Can. J. Math. 32 (1980), 362384.Google Scholar
3. Barucci, V., Anderson, D. F. and Dobbs, D. E., Coherent Mori domains and the principal ideal theorem, Comm. Algebra 15 (1987), 11191156.Google Scholar
4. Gilmer, R., Multiplicative ideal theory (Dekker, New York, 1972).Google Scholar
5. Hungerford, T. W., Algebra (Springer-Verlag, New York, 1974).Google Scholar
6. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, 1966).Google Scholar
7. Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar