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Journal Article

# Some Counterexamples Concerning a Differential Criterion for Flatness

William C. Brown and Sarah Glaz
Proceedings of the American Mathematical Society
Vol. 81, No. 4 (Apr., 1981), pp. 505-510
DOI: 10.2307/2044147
Stable URL: http://www.jstor.org/stable/2044147
Page Count: 6

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Topics: Mathematical rings, Flatness, Differentials, Algebra, Logical proofs

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## Abstract

Let $A$ denote a commutative ring with identity. We suppose $A$ contains a field $k$ of characteristic zero. Let $\Omega^1_k(A)$ and $d: A \rightarrow \Omega^1_k(A)$ denote the $A$-module of first-order $k$-differentials on $A$ and the canonical derivation of $A$ into $\Omega^1_k(A)$ respectively. If $\mathfrak{U}$ is an ideal of $A$ which is flat as an $A$-module, then $xdy - ydx \in \mathfrak{U}^2\Omega^1_k(A)$ for all $x, y$ in $\mathfrak{U}$. We give examples in this paper which show that the converse of this statement is false. We also show that if $\mathfrak{U}$ is a maximal ideal of a Noetherian ring $A$, then $xdy - ydx \in \mathfrak{U}^2\Omega^1_k(A)$ for all $x, y$ in $\mathfrak{U}$ does imply $\mathfrak{U}$ is flat.

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