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A scheme over quasi-prime spectrum of modules

The notions of quasi-prime submodules and developed Zariski topology was introduced by the present authors in [1]. In this paper we use these notions to define a scheme. For an Rmodule M , let X := {Q ∈ qSpec(M) | (Q :R M) ∈ Spec(R)}. It is proved that (X,OX) is a locally ringed space. We study the morphism of locally ringed spaces induced by R-homomorphism M → N , and also by ring homomorphism R → S. Among other results, we show that (X,OX) is a scheme by putting some suitable conditions on M .