Left, right and two-sided ideals

Depends on

• Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
• Subgroup

Used by

• A degenerate one-dimensional Clifford algebra need not be semisimple Counterexample
• A nonradical ideal need not enlarge every tangent space Counterexample
• Quasi-finite does not imply finite Counterexample
• Annihilators, torsion elements and the torsion subset of a module Definition
• Elementary ideals of a finitely presented module Definition
• Hopf ideals and Hopf subalgebras Definition
• Primary decompositions, minimality, and isolated components Definition
• Primary submodules and primary ideals Definition
• Prime ideals and maximal ideals in a commutative ring Definition
• Primitive ideals of an enveloping algebra Definition
• The annihilator of a module over an enveloping algebra Definition
• The associated graded variety of a two-sided ideal Definition
• The central reduction of the enveloping algebra at a central character Definition
• The degenerate affine Hecke algebra Definition
• The degenerate cyclotomic Hecke algebra Definition
• The ideal generated by a subset and principal ideals Definition
• The quotient ring R/I with (r+I)(s+I)=rs+I Definition
• The radical of an ideal Definition
• The submodule IM generated by products of elements of an ideal I with elements of a module M Definition
• The sum I+J and product IJ of two-sided ideals Definition
• The Temperley-Lieb quotient and the Jones specialization Definition
• The type-A finite W-algebra and the Whittaker category Definition
• Universal Clifford algebra of a quadratic space Definition
• Universal enveloping superalgebra Definition
• 2ℤ is an ideal of ℤ but is not a subring under the library's unital convention Example
• Coideals, ring ideals, Hopf ideals, and subbialgebras Example
• Fields and ℤ are Noetherian, and so are their polynomial rings in finitely many variables Example
• Left ideals are exactly the submodules of the regular left module _RR Example
• Primitive ideals of U(sl2) at a generic central character Example
• R×{0} is the kernel of R× S→ S, so (R× S)/(R×{0})≅ S Example
• The dual numbers have Cartan map multiplication by two Example
• The graded dual numbers have Cartan polynomial 1+v² Example
• Vanishing sets and vanishing ideals form a contravariant Galois connection Example
• A field has only the zero ideal and itself, hence is Noetherian Lemma
• All initial forms define the tangent cone Lemma
• Factorization through the Clifford quotient Lemma
• Finite-variable polynomial algebras over fields are Noetherian by finite generators Lemma
• Graded Fitting decomposition for degree-zero endomorphisms Lemma
• Hopf ideals, kernels and quotients of commutative Hopf algebras Lemma
• Ideal criteria and intersections of ideals Lemma

…and 25 more results.

Dependency tree · two levels

9 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)