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Really, this question comes down to listing propeties that are preserved by ring isomorphisms. Off the top of my head, I can think of:

  • cardinality of the ring
  • commutativity
  • the order of elements
  • being a UFD, PID, field etc.

What else is there?

David Lui
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Adil Raza
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  • Related (essentially a duplicate) https://math.stackexchange.com/questions/2039702/what-is-an-homomorphism-isomorphism-saying/2039715#2039715 – Ethan Bolker Dec 27 '23 at 01:58

1 Answers1

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  • Having a multiplicative identity (!)
  • Being free, algebraically closed, real closed, Noetherian, Artinian, local, or (semi)simple
  • Its characteristic, dimension (over the prime field, Krull, projective, injective, global), depth, transcendence degree (when applicable)
  • Having isomorphic $R-Mod$s

Pretty much anything one can think of, in particular literally anything first-order expressible ("Isomorphic structures are a fortiori elementarily equivalent")

ac15
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    The last sentence is the best part of the answer. – Ethan Bolker Dec 27 '23 at 01:56
  • +1: good answer, but you are using "first-order" in an unusually broad sense in your last line: properties like algebraically closed or Noetherian require quantification over sequences or subsets of elements and aren't expressible in the first-order language of the theory of rings. – Rob Arthan Dec 27 '23 at 21:32
  • @RobArthan I'm not, I'm using it in the only possible sense; I don't mean to suggest that all the stuff in the list would fall under such a case: it's just a side comment that adds something if one knows what it's about, and harmless if they don't – ac15 Dec 27 '23 at 21:43
  • OK. Thanks for the clarification. It was just a bit surprising because only your first example is first-order. – Rob Arthan Dec 27 '23 at 22:43
  • aren't "being algebraically/real closed" and "having a given specific characteristic" also first-order expressible? (not finitely of course) – ac15 Dec 27 '23 at 22:57