The q.Zariski topology on the quasi-primary spectrum of a ring

Let $R$ be a commutative ring with identity. We topologize $\mathrm{q.Spec}(R)$, the quasi-primary spectrum of $R$, in a way similar to that of defining the Zariski topology on the prime spectrum of $R$, and investigate the properties of this topological space. Rings whose q.Zariski topology is respectively $T_0$, $T_1$, irreducible or Noetherian are studied, and several characterizations of such rings are given.