In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2025[update].
This question was posed in 1930 by Gottfried Köthe (1905–1989).
Necessary and sufficient conditions under which the statement 3 given above does not hold have been given recently.
"[7] This conjecture, if true, would have proven the Köthe conjecture through the equivalent statements above, however a counterexample was produced by Agata Smoktunowicz.
Partial progress was made when Kelarev[10] produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings.