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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The finiteness of $ I$ when $ {\it R}[{\it X}]/{\it I}$ is flat

Authors: Jack Ohm and David E. Rush
Journal: Trans. Amer. Math. Soc. 171 (1972), 377-408
MSC: Primary 13C05
MathSciNet review: 0306176
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Abstract: Let $ R$ be a commutative ring with identity, let $ X$ be an indeterminate, and let $ I$ be an ideal of the polynomial ring $ R[X]$. Let $ \min I$ denote the set of elements of $ I$ of minimal degree and assume henceforth that $ \min I$ contains a regular element. Then $ R[X]/I$ is a flat $ R$-module implies $ I$ is a finitely generated ideal. Under the additional hypothesis that $ R$ is quasi-local integrally closed, the stronger conclusion that $ I$ is principal holds. (An example shows that the first statement is no longer valid when $ \min I$ does not contain a regular element.)

Let $ c(I)$ denote the content ideal of $ I$, i.e. $ c(I)$ is the ideal of $ R$ generated by the coefficients of the elements of $ I$. A corollary to the above theorem asserts that $ R[X]/I$ is a flat $ R$-module if and only if $ I$ is an invertible ideal of $ R[X]$ and $ c(I) = R$. Moreover, if $ R$ is quasi-local integrally closed, then the following are equivalent: (i) $ R[X]/I$ is a flat $ R$-module; (ii) $ R[X]/I$ is a torsion free $ R$-module and $ c(I) = R$; (iii) $ I$ is principal and $ c(I) = R$.

Let $ \xi $ denote the equivalence class of $ X$ in $ R[X]/I$, and let $ \langle 1,\xi , \cdots ,{\xi ^t}\rangle $ denote the $ R$-module generated by $ 1,\xi , \cdots ,{\xi ^t}$. The following statements are also equivalent: (i) $ \langle 1,\xi , \cdots ,{\xi ^t}\rangle $ is flat for all $ t \geqslant 0$; (ii) $ \langle 1,\xi , \cdots ,{\xi ^t}\rangle $ is flat for some $ t \geqslant 0$ for which $ 1,\xi , \cdots ,{\xi ^t}$ are linearly dependent over $ R$; (iii) $ I = ({f_1}, \cdots ,{f_n}),{f_i} \in \min I$, and $ c(I) = R$; (iv) $ c(\min I) = R$. Moreover, if $ R$ is integrally closed, these are equivalent to $ R[X]/I$ being a flat $ R$-module. A certain symmetry enters in when $ \xi $ is regular in $ R[\xi ]$, and in this case (i)-(iv) are also equivalent to the assertion that $ R[\xi ]$ and $ R[1/\xi ]$ are flat $ R$-modules.

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Keywords: Polynomial ring, flat module, torsion-free module, projective module, content, invertible ideal, finitely generated ideal, principal ideal
Article copyright: © Copyright 1972 American Mathematical Society