TITLE: What are some local properties?
QUESTION [5 upvotes]: What I mean by the question is for example the following statement:
Let $R$ be a ring and suppose that for each prime ideal $\mathfrak{p}$ the local ring $R_{\mathfrak{p}}$ has no nilpotent elements. Then $R$ itself also has no nilpotent elements.
So we call having nilpotent elements a local property. What are other local properties or what are non-local properties? For example being an integral domain is not a local property.

REPLY [3 votes]: Being Cohen-Macaulay is a local property, also by definition.
So is being Gorenstein.
Being universally catenary is local.
Being Noetherian  is not quite local.
Being finitely generated is local.