The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-projective

Brewer, J. W.; Montgomery, P. R.

doi:10.1090/S0002-9939-1973-0323778-8

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 PDF

Proc. Amer. Math. Soc. 41 (1973), 407-414 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

Algèbre commutative

Modules plats

Localisation

36

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. . 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