  Dale Peterson has discovered a surprising result that the quantum cohomology
ring of the flag variety $\mbox{GL}_n(\mathbb{C})/B$ is isomorphic to the
coordinate ring of the intersection of the Peterson variety $\mbox{Pet}_n$ and
the opposite Schubert cell associated with the identity element
$\Omega_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. This is an unpublished result,
so papers of Kostant and Rietsch are referred for this result. An explicit
presentation of the quantum cohomology ring of $\mbox{GL}_n(\mathbb{C})/B$ is
given by Ciocan-Fontanine and Givental-Kim. In this paper we introduce further
quantizations of their presentation so that they reflect the coordinate rings
of the intersections of regular nilpotent Hessenberg varieties
$\mbox{Hess}(N,h)$ and $\Omega_e^\circ$ in $\mbox{GL}_n(\mathbb{C})/B$. In
other words, we generalize the Peterson's statement to regular nilpotent
Hessenberg varieties via the presentation given by Ciocan-Fontanine and
Givental-Kim. As an application of our theorem, we show that the singular locus
of the intersection of some regular nilpotent Hessenberg variety
$\mbox{Hess}(N,h_m)$ and $\Omega_e^\circ$ is the intersection of certain
Schubert variety and $\Omega_e^\circ$ where $h_m=(m,n,\ldots,n)$ for $1<m<n$.
We also see that $\mbox{Hess}(N,h_2) \cap \Omega_e^\circ$ is related with the
cyclic quotient singularity.
