Idempotents in a local ring - Mathematics Stack Exchange
Feb 15, 2015 · 0. Let m m be the unique maximal ideal of a local ring A A. The Jacobson radical (J(R) J (R)) is defined as the intersection of all maximal ideals of A A. Since J(R) J (R) contains the nilradical R R, which is itself an ideal, J(R) = R = m J (R) = R = m by uniqueness. Now, x2 = x x(1 − x) = 0 x 2 = x x (1 − x) = 0;
Can a local ring have more than one prime ideal?
Apr 30, 2015 · 13. A local ring is defined as a ring which has a unique maximal ideal. This unique maximal ideal consists of only non-units and contains all the non-units of the ring R. So examples of local rings include any field, or rings localised at prime ideals and so on.
What is the relationship between local ring and localization?
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.
abstract algebra - Is a subring of a local ring still local ...
Aug 9, 2018 · Every integral domain is a unital subring of a local ring (its field of fractions.) If you know any integral domains that aren't local (pause) then you have your answer. – rschwieb. Aug 9, 2018 at 13:13. I'm not sure whether the author of that website means that a subring of that ring A A is local, or that each subring B B has all elements ...
commutative algebra - Definition of homomorphism of local rings ...
Apr 7, 2021 · 4. Let (R, mR), (S, mS) be two local rings. By definition, a local homomorphism of local rings is a ring homomorphism f: R → S such that the ideal generated by f(mR) in S is contained in mS: f(mR) ⊆ mS. According to Lemma 10.18.3, the condition f(mR) ⊆ mS is equivalent to other conditions.
Localization of a local ring - Mathematics Stack Exchange
Jun 5, 2016 · Now, for the localisation, there is a canonical map A → Am A → A m given by a ↦ a 1 a ↦ a 1. We show that it is an isomorphism by constructing an inverse. Take a b ∈Am a b ∈ A m (technically this element should be written (a, b) (a, b)). Since b ∉m b ∉ m, the inverse b−1 b − 1 exists, and we have by the definition of ...
abstract algebra - Showing $R$ is a local ring if and only if all ...
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abstract algebra - Quotient ring local if Ring is local - Mathematics ...
Jan 5, 2019 · 15. The easiest characterization of local rings here is that a ring is local if and only if it has exactly one maximal ideal. Let M M be the maximal ideal of R R and f: R → R/I f: R → R / I the surjection. If R/I R / I has a maximal ideal M′ ≠ f(M) M ′ ≠ f (M), then f−1(M′) ≠ M f − 1 (M ′) ≠ M is a maximal ideal of R R ...
Local ring of variety at a point - Mathematics Stack Exchange
The local ring really is a "local" property, so yes the local ring of V(x) V (x) at P P is isomorphic to the local ring of V(xy) V (x y) at P P. You might want to try calculating it both ways (see Dustin's comment) and check that you get the same thing. My book's definition: the local ring of V V at a point P P is the ring of f ∈ k(V) f ∈ k ...
What does it mean for $B$ to be a local ring of $A$?
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...