The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$

Depends on

• Left, right and two-sided ideals
• The quotient group $G/N$ and coset product $(gN)(hN)=ghN$
• Every subgroup of an abelian group is normal
• Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides

Used by

• A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
• A scheme is not determined by its underlying topological space Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• homogeneous coordinate ring Definition
• Khovanov–Seidel type A algebra Definition
• Polynomial Weyl algebra and its central parameter Definition
• Quantum Verma modules M(λ), highest weight vectors, and the simple highest weight quotient L(λ) Definition
• Symmetric algebra of a vector space Definition
• The affine scheme of dual numbers Definition
• The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i Definition
• The coordinate ring of a classical affine algebraic set Definition
• The coordinate ring of an affine algebraic set Definition
• The Drinfeld-Jimbo quantized enveloping algebra by generators and relations Definition
• The formal quantum shuffle Borel and its Cartan crossed product Definition
• The HOMFLYPT coefficient ring Definition
• The Khovanov-Seidel path ideal Definition
• The Temperley-Lieb quotient and the Jones specialization Definition
• The triangulated K₀ of the Khovanov–Seidel projective category Definition
• The vertex modules Sᵢ and their prime quotients Definition
• Universal enveloping algebra Definition
• A finite right exact functor needs no infinite-coproduct hypothesis Example
• A finite-dimensional truncated-polynomial Weyl module in characteristic p Example
• A truncated polynomial local ring is Henselian 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
• k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
• Normalization of a nodal affine plane curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• Polynomial differential forms on the line Example
• The dual numbers have Cartan map multiplication by two Example
• The dual-numbers tensor functor is right exact but not left exact Example
• The graded dual numbers have Cartan polynomial 1+v² Example
• All initial forms define the tangent cone Lemma
• Choice-free prime factorisation for a monogenic number ring Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
• Hopf ideals, kernels and quotients of commutative Hopf algebras Lemma
• Lengths of truncated plane local rings Lemma
• Lift a Bezout identity for coprime residue factors Lemma
• Presentation base change and transport of explicit bases to commutative specializations Lemma

…and 22 more results.

Dependency tree · two levels

16 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

• Ernst, An Inquiry-Based Approach to Abstract Algebra, Ideals and Quotient Rings (standard reference, not scraped)