Good morning.
For every module $N$ over a ring $R$, it is defined the grade of $N$ as $j_{R}(N)=\min\left\{i:Ext^{i}_{R}(M,R)\neq0\right\}$. In the book "Zariskian Filtrations" by Li Huishi and F. Van Oystaeyen, page 161, it is defined, for every module $M$, the invariant $J_{s}(M)=\inf\left\{j_{R}(N):N\textrm{ is an }R\textrm{-submodule of }M\right\}$ and then the class $\mathcal{C}_{n}^{R}=\left\{M\textrm{ is an }R\textrm{-module such that }J_{s}(M)\geq n\right\}$. Later, the authors say that this class $\mathcal{C}_{n}^{R}$ is closed by taking submodules, quotients, extensions and direct limits. I obtained the first three ones easily, as the authors suggested, but I am not getting the closure by taking direct limits. I think it should be enough to prove that the class is closed by taking arbitrary direct sums, but still had no results. Thank you.
