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
• Maps from proper integral schemes to the affine line have closed-point image Corollary
• Rings of integers are Dedekind domains Corollary
• The closed points of the prime spectrum are exactly the maximal ideals Corollary
• A common component makes the intersection sum infinite Counterexample
• A nonclosed open immersion is not proper Counterexample
• An intersection of two primitive ideals need not be primitive Counterexample
• Flat and finite type is not open without finite presentation Counterexample
• Frobenius on the affine line is finite flat but not smooth Counterexample
• Quasi-finite does not imply finite Counterexample
• The equation must define the intended scheme Counterexample
• The zero ideal of ℤ is prime but not maximal Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine 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
• A coherent closed-point skyscraper Example
• A finite algebra is its own Zariski Main factor Example
• Computing sqrt((x²,xy)) from its containing primes Example
• Every prime ideal of a product ring comes from one factor Example
• Fitting ideals of a diagonal two-by-two presentation 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
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Empty Proj and irrelevant torsion Lemma

…and 41 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)