Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
First isomorphism theorem for rings:
Statement
First isomorphism theorem for rings: .
Facts & Assumptions
Given: A ring homomorphism .
A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring (A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring).
is an ideal (The kernel of a ring homomorphism is a two-sided ideal).
Ring homomorphisms preserve additive inverses and products (A ring homomorphism satisfies , and for , carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms).
The underlying group map is isomorphic modulo its kernel to its image (First isomorphism theorem for groups: ).
A ring homomorphism preserves (Ring homomorphism: additive, multiplicative, and required to send to ).
Proof
By [L2] and [L1], induces a ring homomorphism with .
This map is surjective by the definition of image, and its additive kernel is trivial, hence it is injective by [L4].
Thus is a ring isomorphism.
Depends on
- A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring
- The kernel of a ring homomorphism is a two-sided ideal
- A ring homomorphism satisfies $f(0) = 0$, $f(-a) = -f(a)$ and $f(ma) = m f(a)$ for $m \in \mathbb{Z}$, carries units to units, and has a subring as its image; composites of ring homomorphisms are ring homomorphisms
- First isomorphism theorem for groups: $G/\ker f\cong\operatorname{im}f$
- Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
Used by
- A product of two Noetherian rings is Noetherian Corollary
- Every algebra of finite type over a Noetherian ring is a Noetherian ring Corollary
- R[x] is Noetherian if and only if R is Noetherian Corollary
- The algebra F[T] generated by an endomorphism is isomorphic to F[x]/(μ_T) Corollary
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras Definition
- R×{0} is the kernel of R× S→ S, so (R× S)/(R×{0})≅ S Example
- A simple algebraic extension is its minimal-polynomial quotient and has power basis 1,a,…,aⁿ⁻¹ and degree n Theorem
- Polynomial functions on an affine algebraic set are its coordinate ring Theorem
- Second isomorphism theorem for rings: S/(S∩ I)≅(S+I)/I Theorem
- Third isomorphism theorem for rings: (R/I)/(J/I)≅ R/J Theorem
Dependency tree · two levels
26 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
- Judson, Abstract Algebra: Theory and Applications, Ring Homomorphisms and Ideals (standard reference, not scraped)