Given a commutative ring $A$ we say that a property P is local if

$A$ has P $\leftrightarrow$ $A_{p}$ has P for all prime ideals $p$ of $A$

It is usually the case that this requirement is equivalent to $A_{m}$ having P for all maximal ideals $m$ of $A$. I was wondering which (if any) are the strongest/most interesting non-local properties $P$ of a commutative ring that do satisfy the second equivalence.

That is to say, what are the most interesting properties P of $A$ such that:

$A_{m}$ has P for all maximal ideals $m$ of $A$ but P is NOT local

The same question applies to any $A$-module $M$.