The finiteness of $I$ when $R[X]/I$ is $R$-projective
HTML articles powered by AMS MathViewer
- by J. W. Brewer and P. R. Montgomery
- Proc. Amer. Math. Soc. 41 (1973), 407-414
- DOI: https://doi.org/10.1090/S0002-9939-1973-0323778-8
- PDF | Request permission
Abstract:
This paper is concerned with the relationship between $R[X]/I$ being a projective $R$-module and $I$ being a finitely generated ideal of $R[X]$. It is shown that if $R[X]/I$ is $R$-free, then $I = fR[X],f$ a monic polynomial of $R[X]$. Also, $R[X]/I$ is a finitely generated projective $R$-module if and only if $R[X]/I$ is a finitely generated $R$-module and $I = fR[X]$ for some $f \in R[X]$. When $R[X]/I$ is projective, $I$ is a finitely generated ideal if and only if $I$ is a principal ideal. Finally, an example is given to show that $R[X]/I$ can be projective without $I$ being finitely generated.References
- N. Bourbaki, Éléments de mathématique. Fasc. XXVII. Algèbre commutative. Chap. 1: Modules plats. Chap. 2: Localisation, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
- S. H. Cox Jr. and R. L. Pendleton, Rings for which certain flat modules are projective, Trans. Amer. Math. Soc. 150 (1970), 139–156. MR 262296, DOI 10.1090/S0002-9947-1970-0262296-4
- William Heinzer and Jack Ohm, The finiteness of $I$ when $R[X]/I$ is $R$-flat. II, Proc. Amer. Math. Soc. 35 (1972), 1–8. MR 306177, DOI 10.1090/S0002-9939-1972-0306177-3
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- Yôichi Miyashita, Commutative Frobenius algebras generated by a single element, J. Fac. Sci. Hokkaido Univ. Ser. I 21 (1970/71), 166–176. MR 0296066
- Masayoshi Nagata, Flatness of an extension of a commutative ring, J. Math. Kyoto Univ. 9 (1969), 439–448. MR 255530, DOI 10.1215/kjm/1250523905
- Jack Ohm and David E. Rush, The finiteness of $I$ when $\textit {R}[\textit {X}]/\textit {I}$ is flat, Trans. Amer. Math. Soc. 171 (1972), 377–408. MR 306176, DOI 10.1090/S0002-9947-1972-0306176-6
- R. Raphael, Some homological results on certain finite ring extensions, Proc. Amer. Math. Soc. 36 (1972), 331–335. MR 308103, DOI 10.1090/S0002-9939-1972-0308103-X
- I. I. Sahaev, The projectivity of finitely generated flat modules, Sibirsk. Mat. Ž. 6 (1965), 564–573 (Russian). MR 0180582
- Wolmer V. Vasconcelos, Simple flat extensions, J. Algebra 16 (1970), 105–107. MR 265342, DOI 10.1016/0021-8693(70)90043-8
- Wolmer V. Vasconcelos, Simple flat extensions. II, Math. Z. 129 (1972), 157–161. MR 319978, DOI 10.1007/BF01187968
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 407-414
- MSC: Primary 13B25
- DOI: https://doi.org/10.1090/S0002-9939-1973-0323778-8
- MathSciNet review: 0323778