I am currently studying the direct limit of rings and I am stuck in the following question. Please help me.
Let $\{R_{i}\}_{i \in I}$ be a nonempty family of commutative rings with unity and $\langle R_{i}, f_{ij} \rangle$ be a direct system over the directed set $I$. Let $f_{i}: R_{i} \rightarrow \varinjlim R_{i}$ be the natural map for all $i$. If $J$ is an ideal of $\varinjlim R_{i}$, then is it true that $J = \varinjlim f_{i}^{-1}(J)$?
My intuition says that it is true, but I could not figure out how to prove this, or, maybe my intuition is wrong. Please help me.
There is a natural map $f_i^{-1}(J)\to J$ induced by $f_i$ for all $i\in I$. This induces a natural map $\varinjlim f_i^{-1}(J)\to J$. We show this is an isomorphism.
Injectivity: This is clear because both limits are taken w.r.t. the same transition maps.
Surjectivity: Take some $a\in J$ then there is an $i$ and an $x\in R_i$ s.t. $x$ represents $a$. This means that $a=f_i(x)$. This shows $x\in f_i^{-1}(J)$. Thus, the map is surjective.