The finite generation of canonical rings and the validity of the Minimal Model Program (MMP) are central themes in the classification of algebraic varieties. In 1987, Sakai constructed a counterexample to the MMP for $\mathbb Q$-Gorenstein surfaces over the complex numbers. However, his construction relies on analytic methods. In this talk, we present purely algebraic constructions over an algebraically closed field of arbitrary characteristic. Specifically, we first provide a counterexample to the MMP for $\mathbb Q$-Gorenstein surfaces. As an application of this construction, we obtain a normal projective $\mathbb Q$-Gorenstein surface whose canonical ring is not finitely generated.