Zero_Product_with_Proper_Zero_Divisor_is_with_Zero_Divisor

⚠️ 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 (R, +, *) be a ring. Let x be a proper zero divisor. Then if x * y = 0 and y ≠ 0, then y is also a zero divisor.

Why this node was retired

The posted statement is

theorem Zero_Product_with_Proper_Zero_Divisor_is_with_Zero_Divisor {R : Type _} [Ring R] (x y : R) (hx : x ≠ 0) (hy : y ≠ 0) (hxy : x * y = 0) : ∃ z : R, z ≠ 0 ∧ y * z = 0 := by sorry

A left annihilator of y is incorrectly converted to a right annihilator in an arbitrary ring.

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 commutative ring, x≠0 and xy=0 imply yx=0, so y has nonzero right annihilator x. For arbitrary rings conclude that y is a left zero divisor, retaining x*y=0.

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.