The radical of an ideal

Depends on

• Commutative ring
• Left, right and two-sided ideals

Used by

• Strong Nullstellensatz: I(V(I)) equals the radical of I Corollary
• A nonradical ideal need not enlarge every tangent space Counterexample
• The equation must define the intended scheme Counterexample
• Primary decompositions, minimality, and isolated components Definition
• Primary submodules and primary ideals Definition
• Systems of parameters and parameter ideals Definition
• The nilradical and reduced rings Definition
• The cusp retains a doubled tangent line Example
• The rank-one 2 by 2 determinantal cone Example
• A classical zero locus depends only on the generated ideal and its radical Lemma
• A separating prime for an element outside a radical Lemma
• An ideal and its radical have the same zero locus Lemma
• Conductor radical detects every polynomial coefficient Lemma
• Finite local length exactly when no common local branch Lemma
• Multiplicity one characterises smooth points with a unique tangent Lemma
• Primes containing an ideal contain its radical Lemma
• Radical membership via positive powers Lemma
• Radicals and quotient correspondence Lemma
• Radicals commute with localization Lemma
• The radical of an ideal is an ideal Lemma
• Algebraic Zariski Main localization at a quasi-finite prime Theorem
• Intersection multiplicity dominates the product of multiplicities, with equality for separated tangent cones Theorem
• The radical of a primary ideal is prime 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)