Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-14T04:45:53.994Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

References

1. Akiba, T., Remarks on generalized quotient rings, Proc. Japan Acad. 40 (1964), 801–806.Google Scholar
2. Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D -f- M, Michigan Math. J. 20 (1973), 79–95.Google Scholar
3. Beauregard, R. A. and Dobbs, D. E., On a class of Archimedean integral domains, Can. J. Math. 28 (1976), 365–375.Google Scholar
4. Bourbaki, N., Commutative algebra (Addison-Wesley, Reading, Mass., 1972).Google Scholar
5. Brewer, J. W. and Rutter, E. A., D + M constructions with general overrings, Michigan Math. J. 23 (1976), 33–42.Google Scholar
6. Dawson, J. and Dobbs, D. E., On going down in polynomial rings, Can. J. Math. 26 (1974), 177–184.Google Scholar
7. Dobbs, D. E., Divided rings and going down, Pac. J. Math. 67 (1976), 353–363.Google Scholar
8. Dobbs, D. E., Coherence, ascent of going down and pseudo-valuation domains, Houston J. Math. 4 (1978), 551–567.Google Scholar
9. Dobbs, D. E., On the weak global dimension of pseudo-valuation domains, Can. Math. Bull. 21 (1978), 159–164.Google Scholar
10. Dobbs, D. E. and Papick, I. J., When is D -\- M coherent?, Proc. Amer. Math. Soc. 56 (1976), 51–54.Google Scholar
11. Eakin, P. M., Jr., The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278–282.Google Scholar
12. Gilmer, R. W., Multiplicative ideal theory, Queen's Papers in Pure and Applied Alathematics 12 (1968) Queen's University, Kingston, Ontario.Google Scholar
13. Greenberg, B., Coherence of Cartesian squares, J. Algebr. 50 (1978), 12–25.Google Scholar
14. Hedstrom, J. R. and Houston, E. G., Pseudo-valuation domains, Pac. J. Math. 75 (1978), 137–147.Google Scholar
15. Hedstrom, J. R. and Houston, E. G., Pseudo-valuation domains, II, Houston J. Math. 4 (1978), 199–207.Google Scholar
16. McAdam, S., Two conductor theorems, J. Algebr. 23 (1972), 239–240.Google Scholar
17. Ohm, J., Some counterexamples related to integral closure in D[[x]], Trans. Amer. Math. Soc. 122 (1966), 321–333.Google Scholar
18. Papick, I. J., Topologically defined classes of going down rings, Trans. Amer. Math. Soc. 219 (1976), 1–37.Google Scholar
19. Richman, F., Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794–799.Google Scholar
20. Sheldon, P. B., Prime ideals in GCD-domains, Can. J. Math. 26 (1974), 98–107.Google Scholar
21. Vasconcelos, W. V., The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381–386.Google Scholar