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A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring
Statement
A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring.
If , is a ring homomorphism, and , there is a unique ring homomorphism such that .
Facts & Assumptions
Given: A two-sided ideal and a ring homomorphism with .
The additive group of a ring is abelian (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).
A two-sided ideal is an additive subgroup (Left, right and two-sided ideals).
Every subgroup of an abelian group is normal (Every subgroup of an abelian group is normal).
The canonical quotient map is a surjective ring homomorphism (The canonical projection is a surjective ring homomorphism with kernel ).
A group homomorphism killing a normal subgroup factors uniquely through its quotient (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).
A ring homomorphism preserves addition, multiplication, and identity (Ring homomorphism: additive, multiplicative, and required to send to ).
Proof
By [L1]--[L3], is normal in the additive group of ; applying [L5] to the additive homomorphism underlying defines and proves representative independence.
Since [L4] gives , one has , and ; thus is a ring homomorphism.
The factor identity holds by step 1.1, and any ring-homomorphic factor is additive, so the uniqueness in [L5] proves its uniqueness as a ring factor.
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
- 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
- Every nonempty principal open is a classical affine variety Theorem
- First isomorphism theorem for rings: R/ker f congimf Theorem
- Localisation commutes with quotient rings: S⁻¹R/S⁻¹I≅ S̄⁻¹(R/I) Theorem
- Universal property of adjoining a root of an irreducible polynomial Theorem
- Universal property of the enveloping algebra Theorem
- Universal property of the symmetric algebra Theorem
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)