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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The finiteness of $I$ when $\textit {R}[\textit {X}]/\textit {I}$ is flat
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by Jack Ohm and David E. Rush PDF
Trans. Amer. Math. Soc. 171 (1972), 377-408 Request permission

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 (An example shows that the first statement is no longer valid when $\min I$ does not contain a regular element.)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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 171 (1972), 377-408
  • MSC: Primary 13C05
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0306176-6
  • MathSciNet review: 0306176