We introduce a family of ideals $I_{n,\lambda,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ partition $k\leq n$ and an integer $s \geq \ell(\lambda)$. This contains both the Tanisaki $I_\lambda$ $I_{n,k}$ Haglund-Rhoades-Shimozono as special cases. study corresponding quotient rings $R_{n,\lambda,s}$ symmetric group modules. When $n=k$ $s$ is arbitrary, we recover Garsia-Procesi modules, whe...