  We introduce a family of ideals $I_{n,\lambda,s}$ in
$\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ a partition of $k\leq n$ and an
integer $s \geq \ell(\lambda)$. This family contains both the Tanisaki ideals
$I_\lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special
cases. We study the corresponding quotient rings $R_{n,\lambda,s}$ as symmetric
group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi
modules, and when $\lambda=(1^k)$ and $s=k$, we recover the generalized
coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for
$R_{n,\lambda,s}$, unifying the monomial bases studied by Garsia-Procesi and
Haglund-Rhoades-Shimozono, and realize the $S_n$-module structure of
$R_{n,\lambda,s}$ in terms of an action on $(n,\lambda,s)$-ordered set
partitions. We also prove formulas for the Hilbert series and graded Frobenius
characteristic of $R_{n,\lambda,s}$. We then connect our work with
Eisenbud-Saltman rank varieties using results of Weyman. As an application of
our work, we give a monomial basis, Hilbert series formula, and graded
Frobenius characteristic formula for the coordinate ring of the
scheme-theoretic intersection of a rank variety with diagonal matrices.
