A scheme over prime spectrum of modules (Q2839022)

Let \(R\) be a commutative ring with identity and \(M\) be a unital \(R\)--module. The prime spectrum of \(M\), the set of all prime submodules of \(M\) which is denoted by \(\text{Spec}(M)\), forms a topological space under the Zariski topology [\textit{C.-P. Lu}, Houston J. Math. 25, No. 3, 417--432 (1999; Zbl 0979.13005)]. Let \(X=\text{Spec}(M)\), then for every open subset \(U\subseteq X\), one may assign a subring \(\mathcal{O}_X(U)\) of \(\prod_{\mathfrak{p}\in\mathcal{O}(U)}R_{\mathfrak{p}}\), where \(\mathcal{O}(U)=\{(P:_RM) \mid P\in U\}\subseteq\text{Spec}(R)\).NEWLINENEWLINEThe main result of the paper, Theorem 2.10, states that \((X,\mathcal{O}_X)\) is a scheme when \(X\) is a \(T_0\)-space and \(M\) is faithful and primeful \(R\)-module. Moreover, if \(R\) is Noetherian ring, then \((X,\mathcal{O}_X)\) is a Noetherian scheme. It is investigated by the authors that for each \(P\in X\), the stalk \(\mathcal{O}_P\) of the sheaf \(\mathcal{O}_X\) is isomorphic to \(R_{\mathfrak{p}}\), where \(\mathfrak{p}=(P:M)\). If \(M\) is faithful and primeful, then for \(f\in R\), the ring \(\mathcal{O}_X(X_f)\) is isomorphic to \(R_f\), where \(X_f\) is the open subset \(X\setminus V(rM)\).NEWLINENEWLINELet \(N\) be another \(R\)-module and \(\varphi:M\longrightarrow N\) be a homomorphism. Looking for an induced morphism between locally ringed spaces of \(\mathcal{O}_{\text{Spec}(N)}\) and \(\mathcal{O}_X\) is the subject of the other results of the authors. They show that when \(\varphi\) is epimorphism, such an induced map exists. They also discuss the problem for a ring homomorphism \(\Phi:R\longrightarrow S\) and an \(S\)-module \(N\), and prove that \(\Phi\) induces a morphism of locally ringed spaces NEWLINE\[NEWLINE(\text{Spec}(N),\mathcal{O}_{\text{Spec}(N)})\longrightarrow(X,\mathcal{O}_X),NEWLINE\]NEWLINE when \(M\) is primeful, \(X\) is a \(T_0\)-space and \((0:_RM)\subseteq (0:_RN)\).