The conditions of the last proposition are not sufficient for a ring $A$ to be artinian.

Let $A$ be a (non-zero) artinian ring. Consider $\prod _{i\in \mathbb{N}}A$. We know that prime ideals in this ring are of the form $\mathfrak{p}\subseteq A$ in one coordinate, and $A$ in the others. Localize at $\mathfrak{p}\times A\times A\times \dots$ and call this new ring $B$.

This new ring has a finite number of maximal ideals and every prime ideal is maximal, but this new ring is not artinian (take a chain by setting a new coordinate to zero at each step).