So I'm taking a course using Atiyah and MacDonald's Introduction to Commutative Algebra, and we just got to local properties, which were defined as a property of a ring $R$ that holds if and only if it holds for $R_P$ for every prime ideal $P\subset R$. The authors then give several examples of local properties, but all of them turn out that there's a third, weaker condition in "the following are equivalent" statements, that the property hold at the localization at every maximal ideal $M$.
So I asked the obvious question: Are there any local properties that require the stronger condition of being true at the localization of every prime?
I.e., is there a ring and a property such that the property IS local, but just being true at every maximal ideal is not sufficient?
The professor not having an example off hand, I figured I'd ask here!