The n-dimensional quantum torus 𝒪 q ((F�) n) is defined as the associative F-algebra generated by x 1,…, xn together with their inverses satisfying the relations xixj= q ij xjxi, where q=(q ij). We show that the modules that are finitely generated over certain commutative sub-algebras ℬ are ℬ-torsion-free and have finite length. We determine the Gelfand–Kirillov dimensions of simple modules in the case when where K. dim stands for the Krull dimension. In this case, if M is a simple 𝒪 q ((F�) n)-module, then 𝒢𝒦-dim (M)= 1 or where 𝒵 (C) stands for the center of an algebra C. We also show that there always exists a simple F* A-module satisfying the above inequality.