Let \(R\) be a commutative ring. We denote by \(\mathrm{Spec}(R)\) the set of all prime ideals of \(R\) and \(j\)-\(\mathrm{Spec}(R)\) the set of all prime ideals in \(\mathrm{Spec}(R)\) that can be written as the intersection of maximal ideals. \(\mathrm{Spec}(R)\) is called the \emph{spectrum} and \(j\)-\(\mathrm{Spec}(R)\) the \emph{\(j\)-spectrum} of \(R.\) Let \(N\) be an \(R\)-module. The \emph{support} of \(N,\) denoted by \(\mathrm{Supp}_R(N),\) is the set of all \(\mathfrak{p} \in\) \(\mathrm{Spec}(R)\) such that \(N_\mathfrak{p}\neq 0.\) In the present paper, the author prove the following results: \begin{itemize} \item[1.] Let \(R\) be a commutative Noetherian ring and \(S\) a possibly noncommutative module-finite \(R\)-algebra. Let \(M\) and \(N\) be finitely generated right \(S\)-modules. Suppose that \(X=j-\mathrm{Spec}(R)\cap \hbox{Supp}_R(N),\) has finite dimension. \begin{itemize} \item[(i)] If \(M_\mathfrak{p}\) maps onto \(N_\mathfrak{p}^{\oplus[t+\dim_X(\mathfrak{p})]}\) for every \(\mathfrak{p}\in X,\) then \(M\) maps onto \(N^{\oplus t}\) \item[(ii)] If \(M_\mathfrak{p}\) admits \(N_\mathfrak{p}^{\oplus[t+\dim_X(\mathfrak{p})]}\) as a direct summand for every \(\mathfrak{p}\in X,\) then \(M\) admits \(N^{\oplus t}\) as a direct summand. \end{itemize} \item[2.] Let \(R\) be a commutative ring and \(S\) a possibly noncommutative module-finite \(R\)-algebra. Let \(M\) be a direct summand of a direct sum of finitely presented right \(S\)-modules, and let \(N\) be a right \(S\)-module. Suppose that \(Y=\mathrm{Max}(R)\cap\hbox{Supp}_R(N),\) is Noetherian of finite dimension \(d.\) \begin{itemize} \item[(i)] Suppose that \(N\) is finitely generated over \(S\). If \(M_\mathfrak{m}\) maps onto \(N_\mathfrak{m}^{\oplus(t+d)}\) for every \(\mathfrak{m}\in Y\), then \(M\) maps onto \(N^{\oplus t}\) \item[(ii)] Suppose that \(N\) is finitely presented over \(S\). If \(M_\mathfrak{m}\) admits \(N_\mathfrak{m}^{\oplus(t+d)}\) as a direct summand for every \(\mathfrak{m}\in Y\), then \(M\) admits \(N^{\oplus t}\) as a direct summand. \end{itemize} \end{itemize}