Math Conjectures

Köthe Conjecture

OPENMajorConjectureProposed 1930 · Full conjecture

Canonical statement

For every associative ring RR (not necessarily unital), if I,J⊆RI,J\subseteq R are left ideals such that every element of II and every element of JJ is nilpotent, then every element of I+JI+J is nilpotent.
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For every associative ring \(R\) (not necessarily unital), if \(I,J\subseteq R\) are left ideals such that every element of \(I\) and every element of \(J\) is nilpotent, then every element of \(I+J\) is nilpotent.
ALG-RING-007:[Kothe1930][Lam2001Noncommutative][Smoktunowicz2001Kothe][Pandey2025Kothe]
Algebra and representation theory

Notes

The Köthe conjecture, raised in Köthe's 1930 study of ring structure, asks whether nilpotence of elements survives the addition of ideals: if II and JJ are left ideals of an associative ring RR, each consisting entirely of nilpotent elements, must every element of I+JI+J be nilpotent [Kothe1930]? Equivalently, is the sum of two nil left ideals again nil, so that every ring has a largest nil one-sided ideal?

The problem admits a web of equivalent reformulations and has been verified for substantial ring classes, including rings with one-sided Noetherian conditions and rings satisfying a polynomial identity; standard accounts appear in Lam's textbook [Lam2001Noncommutative]. Smoktunowicz's work has clarified the boundary of what current techniques give, establishing results closely related to the conjecture and its polynomial-ring reformulations [Smoktunowicz2001Kothe], and the surrounding literature continues to be surveyed [Pandey2025Kothe].

For arbitrary rings the question is unresolved: it is not known that the sum of two nil one-sided ideals is nil, and either a proof in full generality or a counterexample would resolve one of the oldest open problems of noncommutative ring theory.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.