Is the direct limit of Noetherian rings necessarily Noetherian? And if it is, how to prove this? If it is not, what is a counterexample?
(I was thinking this question: if $A_{m}$ are Noetherian for $m\in \mathbb{N}$, with $A_{m}\subseteq A_{m+1}$ is then $\bigcup_{m \in \mathbb{N}}A_m$ necessarily Noetherian?)
The problem in the bracket is homework (if taken $A_{m}$to be $K_{m}[[x_1,...,x_d]]$, where $K_m \subseteq K_{m+1 }$ as proper field extensions), but the general one is not, so if it is wrong, then I would think of other ways to prove or disprove my homework problem.
And please don't tell me the answer to the specific problem. I don't want to spoil it.
A directed colimit (you don't mean limit) of Noetherian ring is usually not Noetherian. For instance, look at $\mathrm{colim}_{n \in \mathbb{N}} k[x_1,\dotsc,x_n] = k[x_1,x_2,\dotsc]$.
But if $(k_n)_{n \in I}$ is a directed system of fields, then $\mathrm{colim}_n k_n$ is also a field, and it follows that also $\mathrm{colim}_n \, k_n[x_1,\dotsc,x_d] = (\mathrm{colim}_n k_n)[x_1,\dotsc,x_d]$ is Noetherian. The last equation does not hold for formal power series rings. Although $(\mathrm{colim}_n k_n)[[x_1,\dotsc,x_d]]$ is Noetherian, I doubt that the subring $\mathrm{colim}_n \, k_n[[x_1,\dotsc,x_d]]$ is always Noetherian.