Commutative ring as a direct limit of Noetherian rings

Question

Does there exist a directed set $\mathcal J$ such that every commutative ring with unity is a direct limit of a family of (commutative) Noetherian rings indexed by $\mathcal J$ ?

Answer 1

No. For instance, consider the ring $R=\mathbb{Z}[S]$ where $S$ is an arbitrary infinite set of indeterminates. Then any Noetherian subring $R_0\subset R$ can only contain finitely many elements of $S$ (otherwise the ideal generated by $R_0\cap S$ would not be finitely generated). It follows that $R$ cannot be a direct limit of a system of Noetherian rings indexed by $\mathcal{J}$ for any $\mathcal{J}$ of cardinality less than $|S|$, and so no fixed $\mathcal{J}$ can work for all $S$.