Prime ideals and maximal ideals in a commutative ring

Depends on

• Left, right and two-sided ideals
• Commutative ring
• Maximal element and greatest element

Used by

• A Noetherian local domain has dimension zero exactly when it is a field Corollary
• For a field K, K⟦ x⟧ is a domain and its nonunits form the unique maximal ideal xK⟦ x⟧ Corollary
• The closed points of the prime spectrum are exactly the maximal ideals Corollary
• The zero ideal of ℤ is prime but not maximal Counterexample
• A local ring is a nonzero commutative ring with a unique maximal ideal Definition
• Associated primes of a module Definition
• Boolean algebra and Boolean ultrafilter Definition
• Jacobson radical and semisimple commutative Banach algebra Definition
• Krull dimension of a nonzero ring Definition
• Localisation at a prime ideal: Rₚ=(R setminusp)⁻¹R Definition
• The height of a prime ideal Definition
• The Jacobson radical of a ring Definition
• The prime spectrum and vanishing sets Definition
• Computing sqrt((x²,xy)) from its containing primes Example
• Every prime ideal of a product ring comes from one factor Example
• Localizing a PID at a nonzero prime Example
• Minimal and maximal primes of the node ring Example
• Null rational sequences form a maximal ideal in the ring of rational Cauchy sequences Example
• The spectrum of a field is a one-point affine scheme Example
• The spectrum of a product ring is a disjoint union Example
• The zero ring has empty spectrum Example
• A prime containing an ideal and avoiding a multiplicative set Lemma
• An ideal contained in a finite union of prime ideals lies in one of them Lemma
• An ideal maximal among the non-finitely-generated ideals is prime Lemma
• An idempotent partitions the spectrum into complementary clopen subsets Lemma
• Every irreducible element of a principal ideal domain is prime Lemma
• In a domain, every prime chain below a prime begins at (0) Lemma
• Maximal ideals of C(X) and zero set ultrafilters Lemma
• Minimal primes over a proper ideal exist Lemma
• Primes containing an ideal contain its radical Lemma
• Primes of a quotient lie over the kernel Lemma
• The Noetherian minimal-prime induction split Lemma
• The spectrum map respects composition and identities Lemma
• The symbolic-power step inside the principal ideal theorem Lemma
• Vanishing sets of finite products Lemma
• A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point Theorem
• Assuming the Axiom of Choice, an element lies in the Jacobson radical exactly when one minus any multiple is a unit Theorem
• Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals Theorem
• Cohen's criterion: a commutative ring in which every prime ideal is finitely generated is Noetherian Theorem
• For a nonconstant p in F[x], the ideal (p) is maximal and F[x]/(p) is a field exactly when p is irreducible Theorem

…and 6 more results.

Dependency tree · two levels

7 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

• Ernst, An Inquiry-Based Approach to Abstract Algebra, Maximal and Prime Ideals (standard reference, not scraped)