primary zariski topology on the primary spectrum of a module

Let r be a commutative ring with identity and let m be an r-module. we define the primary spectrum of m, denoted by ps(m), to be the set of all primary submodules q of m such that (radq : m) = √ (q : m). in this paper, we topologize ps(m) with a topology having the zariski topology on the prime spectrum spec(m) as a subspace topology. we investigate compactness and irreducibility of this topological space and provide some conditions under which ps(m) is a spectral space.