Krull Dimension of Direct Limits of Zero-Dimensional Rings

Question

Is it true that direct limit of a directed system of zero-dimensional rings is zero-dimensional (in the sense of Krull)? Thanks for any help!

If this is true, it is inferred that any prime ideal of the direct limit must be maximal.

Answer 1

If $P\subseteq Q$ are prime ideals in a ring $R=\varinjlim R_i$ with $\dim R_i=0$ for all $i$, then $P\cap R_i=Q\cap R_i$ for all $i$, so $P=Q$.