Let R be the comlimit $\operatorname {lim} R_i$ of rings $R_i$. Let $R_{ired}$ denote the quotient ring $R_i/I_{nil}$ where $I_{nil}$ is the ideal of nilpotent elements. What is $\operatorname {lim} R_{ired}$? Is it $R_{red}$?
Here all rings are commutative with unity.