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Lying-Over Pairs of Commutative Rings

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
University of Tennessee, Knoxville, Tennessee
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(R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 454 - 475
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Anderson, D. F. and Dobbs, D. E., Pairs of rings with the same prime ideals, Can. J. Math. 32 (1980), 362–384.Google Scholar
2. Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D-j-M, Mich. Math. J. 20 (1973), 79–95.Google Scholar
3. Bourbaki, N., Commutative algebra (Addison-Wesley, Reading, Mass., 1972).Google Scholar
4. Chevalley, C., La notion d'anneau de décomposition, Nagoya Math. J. 7 (1954), 21–33.Google Scholar
5. Davis, E. D., Overrings of commutative rings, III: Normal pairs, Trans. Amer. Math. Soc. 182 (1973), 175–185.Google Scholar
6. Dechéne, L. I., Adjacent extensions of rings, dissertation, Univ. of California, Riverside (1978).Google Scholar
7. Dobbs, D. E., On going-down for simple overrings, II, Comm. in Algebra 1 (1974), 439–458.Google Scholar
8. Dobbs, D. E., Ascent and descent of going-down rings for integral extensions, Bull. Austr. Math. Soc. 15 (1976), 253–264.Google Scholar
9. Dobbs, D. E., Coherence, ascent of going-down and pseudovaluation domains, Houston J. Math. 4 (1978), 551–567.Google Scholar
10. Dobbs, D. E., On INC-extensions and polynomials with unit content, Can. Math. Bull. 23 (1980), 37–42.Google Scholar
11. Dobbs, D. E. and Papick, I. J., On going-down for simple overrings, III, Proc. Amer. Math. Soc. 54 (1976), 35–38.Google Scholar
12. Dobbs, D. E. and Papick, I. J., When is D + M coherent?, Proc. Amer. Math. Soc. 56 (1976), 51–54.Google Scholar
13. Dobbs, D. E. and Papick, I. J., Going-down: a survey, Nieuw Archief voor Wiskunde 26 (1978), 255–291.Google Scholar
14. Evans, E. G., Jr., A generalization of Zariski's main theorem, Proc. Amer. Math. Soc. 26 (1970), 45–48.Google Scholar
15. Ferrand, D. and Olivier, J.-P., Homomorphismes minimaux d'anneaux, J. of Algebra 16 (1970), 461–471.Google Scholar
16. Gilmer, R., Integral domains with Noetherian subrings, Comment. Math. Helv. 45 (1970), 129–134.Google Scholar
17. Gilmer, R., Multiplicative ideal theory (Dekker, New York, 1972).Google Scholar
18. Gilmer, R. and Heinzer, W. J., Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7–2 (1967), 133–150.Google Scholar
19. Gilmer, R. and Hoffmann, J. F., A characterization of Priifer domains in terms of polynomials, Pac. J. Math. 60 (1975), 81–85.Google Scholar
20. Hedstrom, J. and Houston, E. G., Pseudovaluation domains, Pac. J. Math. 75 (1978), 137–147.Google Scholar
21. Kaplansky, I., Going up in polynomial rings, unpublished manuscript (1972).Google Scholar
22. Kaplansky, I., Commutative Rings, rev. edition (Univ. of Chicago Press, Chicago, 1974).Google Scholar
23. McAdam, S., Two conductor theorems, J. of Algebra 23 (1972), 239–240.Google Scholar
24. Modica, M. L., Maximal subrings, dissertation, Univ. of Chicago, Chicago (1975).Google Scholar
25. Papick, I. J., Topologically defined classes of going-down rings, Trans. Amer. Math. Soc. 219 (1976), 1–37.Google Scholar
26. Papick, I. J., Local minimal overrings, Can. J. Math. 28 (1976), 788–792.Google Scholar
27. Papick, I. J., Coherent overrings, Can. Math. Bull. 22 (1979), 331–337.Google Scholar
28. Papick, I. J., When coherent rings are Noetherian pairs, Houston J. Math., to appear.Google Scholar
29. Richman, F., Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794–799.Google Scholar
30. Seidenberg, A., A note on the dimension theory of rings, Pac. J. Math. 3 (1953), 505–512.Google Scholar
31. Wadsworth, A., Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc. 195 (1974), 201–211.Google Scholar
32. Zaks, A., A-subalgebras of A[x], J. of Algebra 56 (1979), 287–314.Google Scholar