= Exterior differential ideal
{title2=$d\mathcal I\subseteq\mathcal I$}
An exterior differential ideal on a <smooth manifold> is a graded <ideal> $\mathcal I$ in the <exterior algebra> $\Omega^*(M)$ of <differential forms> that is closed under the <exterior derivative>:
$$
\alpha\in\mathcal I\Longrightarrow d\alpha\in\mathcal I.
$$
Being an ideal also requires $\alpha\wedge\beta\in\mathcal I$ for every $\beta\in\Omega^*(M)$. Thus stability under differentiation is an additional condition, not a consequence of wedge closure.
For example, the ideal generated by $dz$ in $\mathbb R^3$ is differential because $d(f\,dz)=df\wedge dz$. Its integral surfaces include the planes $z=\text{constant}$. In contrast, the ideal generated by $\alpha=dz-x\,dy$ is not differential: $d\alpha=-dx\wedge dy$ and $\alpha\wedge d\alpha=-dz\wedge dx\wedge dy\ne0$, whereas wedging $\alpha$ with any element of its generated ideal gives zero. The <Frobenius theorem> interprets this obstruction as failure of the associated tangent-plane distribution to be integrable.
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