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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Some homological results on certain finite ring extensions


Author: R. Raphael
Journal: Proc. Amer. Math. Soc. 36 (1972), 331-335
MSC: Primary 13B99
MathSciNet review: 0308103
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Abstract: All rings are commutative with identity and all modules are unitary. A ring R is connected if 0 and 1 are the only idempotent elements of R. R is semiconnected if the number of idempotents in R is finite.

Suppose that R is connected, that I is a principal ideal of $ R[x]$, and that $ R[x]/I$ is a finitely generated R-module. Then $ R[x]/I$ is a free R-module.

Suppose that R is semiconnected, that I is a principal ideal of $ R[x]$, and that $ R[x]/I$ is a finitely generated R-module. Then $ R[x]/I$ is a projective R-module.

These results are applied to integral extensions.


References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathématiques. Fasc. XXVII: Algèbre commutative. Chap. 1: Modules plats. Chap. 2: Localization, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
  • [2] Joachim Lambek, Lectures on rings and modules, With an appendix by Ian G. Connell, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0206032 (34 #5857)
  • [3] Masayoshi Nagata, Flatness of an extension of a commutative ring, J. Math. Kyoto Univ. 9 (1969), 439–448. MR 0255530 (41 #191)
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0308103-X
PII: S 0002-9939(1972)0308103-X
Keywords: Flat, projective, free, integral
Article copyright: © Copyright 1972 American Mathematical Society