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 \cite{ah۱۰}. In this paper we use these notions to define a scheme. For an R-module M, let X:=\{Q\in q\Spec(M) \mid (Q:_R M)\in\Spec(R)\}. It is proved that (X, \mathcal{O}_X) is a locally ringed space. We study the morphism of locally ringed spaces induced by R-homomorphism M\rightarrow N, and also by ring homomorphism R\rightarrow S. Among other results, we show that (X, \mathcal{O}_X) is a scheme by putting some suitable conditions on M.

Keywords:

Quasi , Quasi-prime submodule , prime submodule , quasi-primeful module , primeful module , quasi-prime-embedding module , developed Zariski topology , prime , embedding module