Kernel_of_Ring_Homomorphism_is_Subring

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Then the kernel of $\phi$ is a subring of $R_1$.

Why this node was retired

The posted statement is

theorem Kernel_of_Ring_Homomorphism_is_Subring {R₁ R₂ : Type _} [CommRing R₁] [CommRing R₂] (f : RingHom R₁ R₂) : ∃ S : Subring R₁, ∀ x, x ∈ S ↔ f x = 0 := by sorry

Mathlib Subring is unital, while a kernel need only be a nonunital subring/ideal.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For a ring homomorphism f:R→S, its kernel is an ideal and a NonUnitalSubring of R; use either object instead of Subring.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.