A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring

Depends on

• Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
• Left, right and two-sided ideals
• Every subgroup of an abelian group is normal
• The canonical projection $R\to R/I$ is a surjective ring homomorphism with kernel $I$
• A homomorphism that kills a normal subgroup factors uniquely through the quotient group
• Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$

Used by

• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• A nonradical ideal need not enlarge every tangent space Counterexample
• The equation must define the intended scheme Counterexample
• The central reduction of the enveloping algebra at a central character Definition
• The degenerate affine 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 HOMFLYPT coefficient ring Definition
• A finite-dimensional truncated-polynomial Weyl module in characteristic p Example
• A node has two distinct tangent directions Example
• Classical bilinear-form equations linearize to matrix spaces Example
• Different singularities can share a tangent cone Example
• Dual numbers compute the tangent spaces of the general and special linear groups Example
• Normalization of a nodal affine plane curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• The cusp retains a doubled tangent line Example
• The polynomial differential representation and positive-characteristic nonfaithfulness Example
• The product of two parabolas: a block Jacobian and the direct-sum formula Example
• The tangent space of the parabola at a general point and the local parameter Example
• Three axes are not three coplanar lines Example
• A common core with graph-differentiable orbits integrates Heisenberg commutators Lemma
• An augmented coideal ideal of an enveloping algebra is generated by its primitive part Lemma
• Base change of standard smooth presentations Lemma
• Mixed relations, local nilpotence and weights give the integrable quantum action Lemma
• Presentation base change and transport of explicit bases to commutative specializations Lemma
• Standard smooth algebras are finitely presented and flat Lemma
• The coproduct preserves the positive and negative quantum Serre ideals Lemma
• The free associative R-algebra on a set and descent of relations Lemma
• The generic quantum halves form a Drinfeld–Jimbo crossed double Lemma
• The left regular module and its tensor powers detect linear and tensor identities Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• An Artinian ring is canonically the finite product of its localizations at its maximal ideals Theorem
• Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms Theorem
• Commutative rings form a reflective full subcategory of rings Theorem
• Divided-power commutation and the simple U_qᵢ(sl₂) string modules Theorem
• Every nonempty principal open is a classical affine variety Theorem
• First isomorphism theorem for rings: R/ker f congimf Theorem
• Hopf quotients, kernels of Hopf maps, and tensor products Theorem
• Localisation commutes with quotient rings: S⁻¹R/S⁻¹I≅ S̄⁻¹(R/I) Theorem

…and 6 more results.

Dependency tree · two levels

22 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)