TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

How 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: $R / \ker f ≅ im f$

First isomorphism theorem for rings: $R / \ker f ≅ im f$ .

Given: A ring homomorphism $f : R \to S$ .

[L1]

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).

[L2]

[L3]

[L4]

The underlying group map is isomorphic modulo its kernel to its image (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[L5]

technique · direct

1.1L1L2L3L4L5givenconstruct

By [L2] and [L1], $f$ induces a ring homomorphism $\overset{ˉ}{f} : R / \ker f \to im f$ with $\overset{ˉ}{f} (r + \ker f) = f (r)$ .

2.1step 1.1L1L2L3L4L5givenalgebra

This map is surjective by the definition of image, and its additive kernel is trivial, hence it is injective by [L4].

3.1step 2.1∎

Thus $\overset{ˉ}{f}$ is a ring isomorphism.

Depends on