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Algebraic Extensions of Commutative Regular Rings

Published online by Cambridge University Press:  20 November 2018

R. M. Raphael
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In this paper we study algebraic closures for commutative semiprime rings. The main interest, however, is with rings which are regular in the sense of von Neumann. These play the same role with respect to semiprime rings as fields do with respect to integral domains. Two generally distinct notions are defined: “algebraic” and “weak-algebraic” extensions. Each has the transitivity property and yields a closure which is unique up to isomorphism and is “universal”. Both coincide in fields.

The extensions here called “algebraic” were studied independently by Enochs [5] and myself. Our results on these extensions proceed from a different point of view, and allow us to answer a question posed by Enochs. Furthermore, these results are required (and were developed) in order to obtain the weak-algebraic closure, which was the original closure sought. The motivation for the weak-algebraic extensions is found in the work of Shoda [14, p. 134, no. 1].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1133 - 1155
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bourbaki, N., Eléments de mathématique. I: Les structures fondamentales de Vanalyse. Fasc. XI, Livre II: Algèbre: chapitre 4: Polynömes et fractions rationnelles; chaptire 5: Corps commutatifs; deuxième édition, Actualités Sci. Indust., No. 1102 (Hermann, Paris, 1959).Google Scholar
2. Bourbaki, N., Éléments de mathématique, Fasc. XXVII: Algèbre commutative-, chapitre 1: Modules plats; chapitre 2: Localisation; Actualités Sci. Indust., No. 1290 (Hermann, Paris, 1961).Google Scholar
3. Burgess, W., Rings of quotients of group rings, Can. J. Math. 21 (1969), 865–875.Google Scholar
4. Ph., Dwinger, On the completeness of the quotient algebras of a complete Boolean algebra. I, Indag. Math. 20 (1958), 448–456.Google Scholar
5. Enochs, E., Totally integrally closed rings, Proc. Amer. Math. Soc. 19 (1968), 701–706.Google Scholar
6. Faith, C. and Utumi, Y., Intrinsic extensions of rings, Pacific J. Math. 14 (1964), 505–512.Google Scholar
7. Fine, N. J., Gillman, L., and Lambek, J., Rings of quotients of rings of continuous functions (McGill University Press, Montreal, 1965).Google Scholar
8. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, N.J., 1960).Google Scholar
9. Hager, A. W., Isomorphism with a C﹛Y) of the maximal ring of quotients of C(X), Fund. Math. 66 (1969), 7–13.Google Scholar
10. Kaplansky, I., Rings of operators (Benjamin, New York, 1968).Google Scholar
11. Kochen, S., Ultraproducts in the theory of models, Ann. of Math. (2) 74 (1961), 221–261.Google Scholar
12. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham-Toronto-London, 1966).Google Scholar
13. Pierce, R. S., Modules over commutative regular rings, Mem. Amer. Math. Soc, No. 70 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
14. Shoda, K., Zur Théorie der algebraischen Eriveiterungen, Osaka Math. J. 4 (1952), 133–143.Google Scholar
15. Storrer, H., Epimorphismen von kommutativen Ringen, Comment. Math. Helv. 43 (1968), 378–401.Google Scholar
16. Zariski, O. and Samuel, P., Commutative algebra, Vol. I, The University Series in Higher Mathematics (Van Nostrand, Princeton, N.J., 1958).Google Scholar