On the Zariski topology over the primary-like spectrum

Abstract

Let R be a commutative ring with identity and M be a unital R-module. The primary-like spectrum PS(M) has a topology which is a generalization of the Zariski topology on the prime spectrum Spec(R). We get several topological properties of PS(M), mostly for the case when the continuous mapping ϕ : PS(M) → Spec(R/Ann(M)) defined by ϕ(Q) = p (Q : M)/Ann(M) is surjective or injective. For example, if ϕ is surjective, then PS(M) is a connected space if and only if Spec(R/Ann(M)) is a connected space. It is shown that if ϕ is surjective, then a subset Y of PS(M) is irreducible if and only if Y is the closure of a singleton set. It is also proved that if the image of ϕ is a closed subset of Spec(R/Ann(M)), then PS(M) is a spectral space if and only if ϕ is injective.