Köthe's conjecture asserts that the sum of two nil left ideals of a ring is nil. Here an ideal is nil when each of its elements is nilpotent, with the exponent allowed to depend on the element. This is weaker than requiring the ideal itself to be nilpotent. The analogous assertion for right ideals is equivalent.
An equivalent matrix formulation asserts that 
is nil whenever 
is a nil ring. This illustrates the
difficulty: nilpotence of each matrix entry does not by itself force nilpotence of
the matrix. The equivalence and the original conjecture
are discussed by Adamczewski (2026).
Adamczewski, Böhmler, and Marczinzik (2026) gave a counterexample over any countable field 
. They construct a nil 
-algebra 
generated by 
and 
such that the matrix 
is not nilpotent. In the unitization of 
, this gives two nil left
ideals of the matrix ring whose sum is not nil, refuting
the original formulation. The initial construction was produced by GPT-6 Astra and
formalized in Lean; the subsequent paper develops the ring-theoretic
argument and its connection with the original conjecture
(Adamczewski 2026; Adamczewski et al. 2026).
