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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 local properties $P$ of a commutative ring that do not satisfy the second equivalence. Similarly, I would like to know the strongest/most interesting non-local properties P that are true at all localizations at $p$.

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

(1) $A_{m}$ A_{p}$has P for all maximal prime ideals$m$p$ of $A$ but P is NOT local

The same question applies to any

or

(2) P is local BUT it is NOT true that if $A$-module A_m$has P for all maximal ideals$M$.m$ of $A$ then $A$ has P.

EDIT: After comments and answers received have edited (and expanded) the question. Hope it is clear and unambiguous now.

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# What is the "strongest" non-local property of a ring/module that is true of all localizations at maximal ideals?

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 $M$.