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The localization construction is extremely useful in algebraic geometry.

But this object seems for me very natural (of course, that's maybe only a mistake of my immature mind) for commutative rings itself as a way to describe the properties of rings like $R[x]/\langle ax-1\rangle$ and some more complicated commutative $R$-algebras.

As an example of straightforward applications of the concept in ring theory I could remember at least the least tricky way to prove the description of a nilradical:

For a commutative ring $R$ the following holds: $$\sqrt{\langle 0\rangle}=\bigcap_{I\in \operatorname{Spec} R} I.$$

But are there any more applications of a localization in ring theory. Maybe ones that don't use other constructions?

J. W. Tanner
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Mitya Kustov
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    There are many applications in commutative algebra (going up theorem, going down theorem etc., see for example here) and algebraic number theory in general. – Dietrich Burde Feb 11 '23 at 21:25
  • See e.g. What are some local properties? and similar questions. – Bill Dubuque Feb 11 '23 at 22:01
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    Localization is the basic technique of modern commutative ring theory, it is very hard to pick a book on the subject and not find localization in some form or another. And a good part of noncommutative ring theory consists of lamenting that one cannot localize and is forced to do weird things, so one could count that also as an application. You question is best answered by suggesting you pick any textbook on commutative algebra! – Mariano Suárez-Álvarez Feb 12 '23 at 01:02

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A very tangible use in basic algebraic number theory is to localize "at" a prime to study primes lying over it... because a Dedekind ring with a single prime (as is the localized version) is necessarily a principal ideal domain, and many arguments become (thereafter) essentially elementary-intuitive. :)

paul garrett
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