Aug 19, 2018 · As I am learning localization, it is easy to see that the localization of a ring $ R $ at $ p $, where $ p $ is a prime ideal of $ R $, is a local ring. And I also know that the localization of a local ring is actually isomorphic to itself hence also a local ring.

May 2, 2015 · In algebraic-geometric or commutative-algebraic contexts, "local ring" usually also means Noetherian. Share.

Jun 5, 2016 · Localization of regular local ring by prime ideal. 1. Question about localization of a local ring. 0.

Jan 2, 2022 · Which seems like it would make sense in a context where local rings actually are rings of germs: local rings with trivial maximal ideal (fields) are germs of functions on a trivial space. But how to generalize this?

May 2, 2023 · In Atiyah and Macdonald, an exercise states: Let A be a valuation ring of a field K. Show that every subring of K which contains A is a local ring of A. I tried looking up what this means and found...

Mar 24, 2020 · $\begingroup$ @user100101212 It's well known that localizing at any prime ideal ($(x,y)$ would qualify) results in a local ring. $\endgroup$ – rschwieb Commented Mar 24, 2020 at 20:00

Example of a ring with a unique two sided maximal ideal which is not a local ring (that is it has more than one left or right maximal ideals). 1 Simple rings/algebras fail to be division rings/algebras because of zero-divisors?

Every element of a local ring is a sum of two units iff the cardinality of the ring is less than twice the cardinality of units Hot Network Questions circuitikz, coloring bipole nodes with different colors

Feb 15, 2015 · Is it true that a local ring, i.e., a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\\neq 0, 1$ ? Why ? Any hint ?

Prove that: a) If A is a local ring then A has characteristic zero or a power prime. Proof. Suppose M is ...