Commutative algebra Basic notions Concept
Ideal
Summary
An ideal of a ring is an additive subgroup that is closed under multiplication by every element of the ring. Ideals are the kernels of ring homomorphisms, and they give quotient rings. The definition of an ideal gives the conditions.
Where this leads
56 pages build on this one.
- Nakayama's lemma
- Regular local rings are UFDs
- Krull intersection theorem
- Algebraic set
- Ideal sheaf
- Étale morphism of rings
- Artin–Rees lemma
- Homogeneous ideal
- Local cohomology
- Gröbner basis
- Codimension of a singularity
- Ideal of an algebraic set
- Rees algebra
- Complete local ring
- Ideal norm
- Flat module
- Fractional ideal
- Finitely generated projective modules over local rings are free
- Linear topology
- Determinant trick
- Hilbert–Burch theorem
- Exterior differential system
- Regular chain
- Tropical variety
- Formally smooth map
- Hilbert's Nullstellensatz
- Coordinate ring
- Finite determinacy theorem
- Positivstellensatz
- Filtration of a module
- Milnor number
- Quasi-compactness of the spectrum
- Determinantal ideal
- I-adic topology
- Baer's criterion
- Algebraic statistics
- Depth
- Completion is flat
- Completion of a ring
- Mather–Yau theorem
- Dedekind zeta function
- Tjurina number
- Unique factorization of ideals in a Dedekind domain
- Cayley–Hamilton theorem for modules
- Invertible ideal
- Elimination theory
- Regular local rings are domains
- Quasi-compact morphism
- Briançon–Skoda theorem
- Associated graded ring
- Tight closure
- Affine variety
- A Dedekind UFD is a PID
- Monomial ideal
- Colombeau algebra
- Weak Nullstellensatz
Further reading
- Web https://en.wikipedia.org/wiki/Ideal_(ring_theory), lead section
About this page
| Tag | 096Z9MP2ME |
|---|---|
| URI | m2://096Z9MP2ME |
| Kind | concept |
| Writing | W1 stated |
| MSC | 13A15 |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
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What links here
Special cases
- Prime ideal
- Ideal generated by a subset
- Principal ideal
- Homogeneous ideal
- Irreducible ideal
- Primary ideal
Definitions
Used by
- Nakayama's lemma
- Krull intersection theorem
- Algebraic set
- Ideal sheaf
- Étale morphism of rings
- Artin–Rees lemma
- Homogeneous ideal
- Local cohomology
- Gröbner basis
- Codimension of a singularity
- Ideal of an algebraic set
- Rees algebra
- Ideal norm
- Flat module
- Fractional ideal
- Linear topology
- Determinant trick
- Hilbert–Burch theorem
- Exterior differential system
- Regular chain
- Tropical variety
- Formally smooth map
- Hilbert's Nullstellensatz
- Finite determinacy theorem
- Filtration of a module
- Milnor number
- Quasi-compactness of the spectrum
- Determinantal ideal
- I-adic topology
- Baer's criterion
- Algebraic statistics
- Depth
- Completion is flat
- Completion of a ring
- Mather–Yau theorem
- Dedekind zeta function
- Tjurina number
- Unique factorization of ideals in a Dedekind domain
- Cayley–Hamilton theorem for modules
- Invertible ideal
- Elimination theory
- Briançon–Skoda theorem
- Associated graded ring
- Tight closure
- Monomial ideal
- Colombeau algebra
- Weak Nullstellensatz