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The nilradical is the intersection of all prime ideals
Statement
Assume the Axiom of Choice.
For a commutative ring , , with the empty-intersection convention in force for the zero ring.
Facts & Assumptions
Given: A commutative ring and the Axiom of Choice.
The nilradical of is (The nilradical and reduced rings).
The radical of any ideal is the intersection of the prime ideals containing it (The radical of an ideal is the intersection of the prime ideals containing it).
Proof
By [L1], .
Applying [L2] to the zero ideal gives . Combining this with step 1.1 yields the claimed formula for the nilradical.
Therefore the nilradical is exactly the intersection of all prime ideals of .
Depends on
Used by
- A ring is reduced exactly when zero is an intersection of primes Corollary
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- The prime spectrum is connected exactly when the ring has no idempotents other than zero and one Corollary
- A clopen decomposition of the spectrum comes from a nontrivial idempotent Lemma
- Closed immersions are affine quotients and survive base change Lemma
- Distinguished-subset identities Lemma
- Finite-type field extensions with zero Ω Lemma
- Localization sections are independent of a distinguished-open presentation Lemma
- Strong transcendence descends to reduced minimal-prime quotients Lemma
- A Noetherian ring is Artinian exactly when every prime ideal is maximal Theorem
- An Artinian local ring has nilpotent maximal ideal, and its finite modules have finite length Theorem
- An Artinian ring is canonically the finite product of its localizations at its maximal ideals Theorem
- Closed immersions into affine schemes are quotient spectra Theorem
- Every commutative Artinian ring is Noetherian Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, A Primer of Commutative Algebra, v4.03, §2 Ideals (standard reference, not scraped)
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §2 Ideals (standard reference, not scraped)