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

• A nonclosed open immersion is not proper Counterexample
• A nonradical ideal need not enlarge every tangent space Counterexample
• For a field F, the ideal (x,y) in F[x,y] is not principal Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The equation must define the intended scheme Counterexample
• Determinantal divisors from the minors of a matrix over a PID Definition
• Elementary ideals of a finitely presented module Definition
• Finitely presented modules and finitely presented algebras Definition
• Polynomial Weyl algebra and its central parameter 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 degenerate affine Hecke algebra Definition
• The degenerate cyclotomic Hecke algebra Definition
• The Drinfeld-Jimbo quantized enveloping algebra by generators and relations Definition
• The formal quantum shuffle Borel and its Cartan crossed product Definition
• The Khovanov-Seidel path ideal Definition
• The monic greatest common divisor of two polynomials over a field Definition
• The prime spectrum and vanishing sets Definition
• The type-A finite W-algebra and the Whittaker category Definition
• Universal enveloping algebra Definition
• Universal enveloping superalgebra Definition
• A finite-dimensional truncated-polynomial Weyl module in characteristic p Example
• A node has two distinct tangent directions Example
• Class group of Q(sqrt -5) Example
• Class group of Q(sqrt 10) Example
• Different singularities can share a tangent cone Example
• Dual numbers compute the tangent spaces of the general and special linear groups Example
• Explicitly lifting the two simple labels in a formal rank-two algebra Example
• 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
• Polynomial differential forms on the line Example
• The cusp retains a doubled tangent line Example
• The polynomial differential representation and positive-characteristic nonfaithfulness Example
• The polynomial ring in countably many variables is not Noetherian Example
• The product of two parabolas: a block Jacobian and the direct-sum formula 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
• The tangent space of the parabola at a general point and the local parameter Example
• Three axes are not three coplanar lines Example

…and 37 more results.

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)