Some counterexamples concerning a differential criterion for flatness

Abstract:

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