Summary: The generalized non-commutative torus \(T_{\varrho}^{k}\) of rank \(n\) is defined by the crossed product \(A_{{m}/{k}} \times_{\alpha_3} {\mathbb Z} \times_{\alpha_4}\dots\times_{\alpha_n} {\mathbb Z}\), where the actions \(\alpha_i\) of \({\mathbb Z}\) on the fibre \(M_k({\mathbb C})\) of a rational rotation algebra \(A_{{m}/{k}}\) are trivial, and \(C^*(k{\mathbb Z} \times k{\mathbb Z}) \times_{\alpha_3} {\mathbb Z} \times_{\alpha_4} \dots \times_{\alpha_n} {\mathbb Z}\) is a non-commutative torus \(A_{\varrho}\). It is shown that \(T^k_{\varrho}\) is strongly Morita equivalent to \(A_{\varrho}\), and that \(T_{\varrho}^{k} \otimes M_{p^{\infty}}\) is isomorphic to \(A_{\varrho} \otimes M_{k}({\mathbb C}) \otimes M_{p^{\infty}}\) if and only if the set of prime factors of \(k\) is a subset of the set of prime factors of \(p\).