Zero_Divisor_Product_is_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.

The ring product of a zero divisor with any ring element is a zero divisor.

Why this node was retired

The posted statement is

theorem Zero_Divisor_Product_is_Zero_Divisor {R : Type _} [Ring R] (x y z : R) (hx : x ≠ 0) (hz : z ≠ 0) (hxz : x * z = 0) : ∃ w : R, w ≠ 0 ∧ (x * y) * w = 0 := by sorry

Right-annihilator closure is asserted under right multiplication in any noncommutative 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 R, if z≠0 and xz=0, then (xy)*z=0 for every y. For arbitrary rings distinguish left and right zero divisors and use a side-correct statement instead.

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.