Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T11:43:47.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Coherent Overrings

Published online by Cambridge University Press:  20 November 2018

Ira J. Papick
Ira J. Papick*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri65201
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.

In the study of particular categories of integral domains, it has proved useful to know which overrings of the domains of interest lie within the category, and indeed whether all such overrings do. (Recall: an overring of R is a ring T with R ⊆ T ⊆ quotient field of R.) Two classes of domains classically studied in this setting are Prüfer domains and one-dimensional Noetherian domains. Since both of these classes are contained in the category of coherent domains, it is natural to investigate this category in this setting.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 3 , 01 September 1979 , pp. 331 - 337
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Atiyah, M. F. and MacDonald, I. G., Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969.Google Scholar
2. Bourbaki, N., Commutative Algebra, Addison-Wesley, Reading, Mass., 1972.Google Scholar
3. Chase, S. U., Direct products of modules, Trans. Amer. Math. Soc., 97 (1960), 457-519.Google Scholar
4. Davis, E. D., Overrings of commutative rings. Ill: Normal Pairs, Trans. Amer. Math. Soc, 182 (1973), 175-185.Google Scholar
5. Davis, E. D., Integrally closed pairs, Lecture Notes in Math., Vol. 311, Springer Verlag, New York, 1970.Google Scholar
6. Dobbs, D. E. and Papick, I. J., When is D + M coherent? Proc. Amer. Math. Soc, 56 (1976), 51-54.Google Scholar
7. Greenberg, B., Coherence in cartesian squares, J. of Algebra, 50 (1978), 12-25.Google Scholar
8. Harris, M. E., Some results on coherent rings, Proc. Amer. Math. Soc, 17 (1966), 474-479.Google Scholar
9. Kaplansky, I., Commutative Rings, Allyn and Bacon, Boston, Mass., 1970.Google Scholar
10. Krull, W., Einbettungsfreie, fast-Noethersche Ringe und ihre oberringe, Math. Nachr., 21 (1960), 319-338.Google Scholar
11. McAdam, S., Two conductor theorems, J. of Algebra, 23 (1972), 239-240.Google Scholar
12. McAdam, S., Simple going down, J. London Math. Soc (2), 13 (1976), 167-173.Google Scholar
13. Papick, I. J., Topologically defined classes of going-down domains, Trans. Amer. Math. Soc, 219 (1976), 1-37.Google Scholar
14. Papick, I. J., A remark on coherent overrings, Can. Math. Bull., 21 (1978), 373-375.Google Scholar
15. Papick, I. J., Finite type extensions and coherence, Pac J. Math., 78 (1978), 161-172.Google Scholar
16. Raynaud, M., Anneaux Locaux Henséliens, Lecture Notes in Math., Vol. 169, Springer Verlag, New York, 1970.Google Scholar
17. Richman, F., Generalized quotient rings, Proc. Amer. Math., Soc, 16 (1965), 794-799.Google Scholar
18. Seidenberg, A., A note on the dimension theory of rings, Pacific J. Math., 3 (1953), 505-512.Google Scholar
19. Wadsworth, A., Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc, 195 (1974), 201-211.Google Scholar