Köthe’s Conjecture (general case)

Determine whether, for an arbitrary ring A with no nonzero nil ideals, A has nonzero one-sided nil ideals; that is, ascertain the general validity of Köthe’s Conjecture across all rings.

Background

The paper reviews the historical context of Köthe’s Problem (often referred to as Köthe’s Conjecture), originally proposed in 1930, and notes that while it has been confirmed in certain classes of rings, a general solution is still lacking. The authors emphasize the relevance of this conjecture to broader themes in ring theory, including connections to Kurosh’s Problem.

Within this work, the authors prove that nil complete metric algebras over normed fields of characteristic zero are nilpotent and deduce a positive answer to Köthe’s Problem for complete metric algebras over normed fields of characteristic zero (Corollary 2.8). Nonetheless, the conjecture remains unresolved in full generality beyond the specific class treated.

References

This last problem was proposed by G. Köthe in 1930 (see [20]), and since then, this conjecture has been confirmed in some classes of rings, but it still does not have a general solution. Köthe's Problem asks 'if a ring A has no nonzero nil ideals, then does " have nonzero one-sided nil ideals?'.

— The Nilpotency of the Nil Metric $\mathbb{F}$-Algebras (2504.17168 - França, 24 Apr 2025) in Section 1 (Introduction)