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


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

Authors: J. W. Brewer and P. R. Montgomery
Journal: Proc. Amer. Math. Soc. 41 (1973), 407-414
MSC: Primary 13B25
MathSciNet review: 0323778
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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.

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PII: S 0002-9939(1973)0323778-8
Keywords: Polynomial ring, flat module, free module, projective module, content, finitely generated ideal, principal ideal
Article copyright: © Copyright 1973 American Mathematical Society