Frobenius-Perron dimensions of integral ${\mathbb{Z}}_{+}$-rings and applications

We introduce the notion of the Frobenius-Perron dimension of an integral ${\mathbb{Z}}_{+}$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension $p$.