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Injective Modules Over Prufer Rings

Published online by Cambridge University Press:  22 January 2016

Eben Matlis
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The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 15 , August 1959 , pp. 57 - 69
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

References

[1] Cartan, H. and Eilenberg, S., Homological Algebra, Princeton University Press, 1956.Google Scholar
[2] Fleischer, I., Modules of finite rank over Prufer rings, Annals of Mathematics, vol. 65, No. 2 (1957), pp. 250–254.CrossRefGoogle Scholar
[3] Hattori, A., On Prufer rings, Journal of the Mathematical Society of Japan, vol. 9, No. 4 (1957), pp. 381–385.CrossRefGoogle Scholar
[4] Kaplansky, I., Modules over Dedekind rings and valuation rings, Transactions of the American Mathematical Society, vol. 72, No. 2 (1952), pp. 327–340.CrossRefGoogle Scholar
[5] Kaplansky, I., Dual modules over a valuation ring I, Proceedings of the American Mathematical Society, vol. 4, No. 2 (1953), pp. 213–217.CrossRefGoogle Scholar
[6] Matlis, E., Injective modules over Noetherian rings, Pacific Journal of Mathematics, vol. 8, No. 3 (1958), pp. 511–528.CrossRefGoogle Scholar
[7] Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Science Reports of the Tokyo Kyoiku Daigaku, Sect. A, vol. 6, No. 150 (1958), pp. 83–142.Google Scholar
[8] Zelinsky, D., Linearly compact modules and rings, American Journal of Mathematics, vol. 75, No. 1 (1953), pp. 79–90.Google Scholar