The ideal generated by a subset and principal ideals

Depends on

• Left, right and two-sided ideals
• Ideal criteria and intersections of ideals

Used by

• For a field F, the ideal (x,y) in F[x,y] is not principal Counterexample
• Determinantal divisors from the minors of a matrix over a PID Definition
• Finitely presented modules and finitely presented algebras Definition
• Principal ideal domain Definition
• Symmetric algebra of a vector space Definition
• The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial Definition
• The monic greatest common divisor of two polynomials over a field Definition
• The prime spectrum and vanishing sets Definition
• Universal enveloping algebra Definition
• Fields and ℤ are Noetherian, and so are their polynomial rings in finitely many variables Example
• k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
• The polynomial ring in countably many variables is not Noetherian Example
• The subalgebra k[x,xy,xy²,…] of k[x,y] is not Noetherian Example
• The subring k[x,y,x/y,x/y²,…] of k(x,y) has a strictly ascending chain of principal ideals Example
• Working the Hilbert basis construction on an ideal of ℤ[x] with non-monic stages Example
• ℤ and k[x] are Noetherian but not Artinian Example
• False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
• FALSE: every finitely generated module over a domain is a direct sum of cyclic modules False statement
• An Artinian integral domain is a field Lemma
• An ideal maximal among the non-finitely-generated ideals is prime Lemma
• Distinguished-subset covers detect radicals Lemma
• Every ideal of a localisation is generated by the images of any generating set of its contraction Lemma
• If some ideal is not finitely generated, there is one maximal among the ideals that are not Lemma
• The spectrum map pulls back vanishing sets Lemma
• A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member Theorem
• Chinese remainder theorem for pairwise comaximal ideals Theorem
• Commutative rings form a reflective full subcategory of rings Theorem
• Every Euclidean domain is a principal ideal domain 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
• In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra Theorem
• The nilradical of an Artinian ring is a nilpotent ideal 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

• Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)