Corollary 2

Let $A$ be a regular noetherian local ring, and $q$ an ideal of $A$. Then $A/q$ is regular if and only if $q$ is generated by a subset of a system of coordinates of $A$.

The condition is sufficient by Prop. 2.

Suppose $A/q$ regular, and let $x = (x_1, ..., x_r)$ be a sequence of elements of $q$ whose classes modulo $m_A^2$ form a basis of $(q + m_A^2)/m_A^2$ over the field $\kappa_A$. Let $x$ be the ideal of $A$ generated by $x$. Then $x \subset q$ and $x$ forms part of a system of coordinates of $A$, hence the noetherian local ring $A/x$ is regular (Prop. 2); moreover, the vector spaces $m_A/(q + m_A^2)$ and $m_A/(x + m_A^2)$ have the same dimension over $\kappa_A$. It follows that the regular noetherian local rings $A/q$ and $A/x$ have the same dimension. Since the ideals $q$ and $x$ are prime and one has $x \subset q$, one concludes finally that $q = x$.