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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.

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  • $\begingroup$ Do you mean limit or colimit? $\endgroup$
    – Martin Brandenburg
    Jul 6, 2013 at 8:51

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If your rings are commutative then the answer (to your second question) is yes. You can find a proof in EGA 0.6.1.3 (second edition).

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  • $\begingroup$ I just want to add more information to Fred's answer for Mohan's first question. To take colimit (I don't know why people call this colimit, I thought it should be called direct limit stacks.math.columbia.edu/tag/0277) of $R_i$, $R_i$ must be a direct system. Hence, you have a composition map $R_i \rightarrow R_j \rightarrow R_j/I_j$, whose kernel contains $I_i$, whenever $i \leq j$. This makes $R_i/I_i$ a directed system and you can take its direct limit (colimit). $\endgroup$
    – mr.bigproblem
    Jul 6, 2013 at 4:56
  • $\begingroup$ It's called colimit instead of direct limit as it is an instance of the notion of colimit of a functor between categories, in which the first is a directed poset. $\endgroup$
    – Edoardo Lanari
    Jul 6, 2013 at 10:11

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