I read about inverse limits in this post, and found the example by Arturo Magidin quite interesting (his "approximate" solution of $x^2 = -1$ in $\mathbb Z$).
By his construction we get a Ring which is an extension of $\mathbb Z$, i.e. the ring of $p$-adic (here $5$-adic) integers where this equation has a solution. On the other side it is well know that $i$ solves this equation, so $\mathbb Z[i] \equiv \mathbb Z[x] / (x^2+1)$ is also a Ring which is an extension of $\mathbb Z$ such that this equation is solvable. Could something be said about the relation on these different ring extensions?
EDIT: Correction from comment $x^2 - 1 \to x^2 + 1$.