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Let $A$ be a ring (commutative with $1$ and $0$). A property $P$ is said to be a local property if the following holds true:

$$A ~\text{has}~ P \iff A_{\mathfrak{p}} ~\text{has}~ P ~\text{for each prime ideal $\mathfrak{p}$ of $A$}$$

What are some examples of local properties? (specifically for rings, not modules)

Jonas Hardt
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  • Reduced, Cohen-Macaulay, Gorenstein.... A very important fact is that being the 0 module is also a local property. – walkar Apr 18 '23 at 01:20
  • Does this answer your question? What are some local properties? – Viktor Vaughn Apr 18 '23 at 01:38
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    @ViktorVaughn Hmm, as written that post seems to be defining "local property" as "if all localizations of $R$ at prime ideals have property P, $R$ has property P." This post specifies it is biconditional. Don't have an example to distinguish, however. Good to link the post anyhow. – rschwieb Apr 18 '23 at 03:47
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    @rschwieb I don't think I have ever encountered a ring-theoretic property which is local-to-global but not global-to-local (in general, I guess any property which passes to subrings and (arbitrary) products is local-to-global) – math54321 Apr 18 '23 at 18:25
  • @math54321 Yeah, nothing comes to mind for me either. – rschwieb Apr 18 '23 at 18:57
  • @rschwieb It just occurred to me that the property "has a nontrivial idempotent" is (vacuously) local-to-global, but not global-to-local (alternatively, any property which is never satisfied for local rings works) – math54321 Apr 18 '23 at 19:12
  • @rschwieb Actually here is a much better example: the property "has trivial unit group" passes to subrings and products, but is not global-to-local. It is also not vacuously false for local rings, as there is a unique(!) local ring that satisfies it – math54321 Apr 18 '23 at 19:36
  • @math54321 Yeah i was just trying to articulate that the vacuous thing wasn't satisfying. Glad you found the new one. – rschwieb Apr 18 '23 at 19:37

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