The notions of quasi-prime submodules and developed  Zariski topology was introduced by the present authors in cite{ah10}. In this paper we use these notions to define a scheme. For an $R$-module $M$, let $X:={Qin qSpec(M) mid (Q:_R M)inSpec(R)}$. 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 $Mrightar...

1 Basic Properties Definition 1. Let X be a scheme. We denote the category of sheaves of OX -modules by OXMod or Mod(X). The full subcategories of qausi-coherent and coherent modules are denoted by Qco(X) and Coh(X) respectively. Mod(X) is a grothendieck abelian category, and it follows from (AC,Lemma 39) and (H, II 5.7) that Qco(X) is an abelian subcategory of Mod(X). If X is noetherian, then ...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.