The Kothe conjecture, in ring theory, is open as of 2025. One way to state it is that if a ring R has no nonzero nil ideal, then it has no nonzero nil one-sided ideal either. It was posed in 1930 by Gottfried Kothe. The conjecture has been shown to hold for various classes of rings, such as polynomial identity rings and right Noetherian rings, but a general solution remains elusive. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Chronology
Resolved YeararXiv preprint posted 2026-09-07, not yet peer reviewed Classification
Resolution StatusarXiv preprint posted 2026-09-07, not yet peer reviewed Prize Status
Prize Status (category) Proposed Year Connections
In Branch
Source Köthe conjecture (Wikipedia)
Sources
1. Kothe Conjecture (Wikipedia)
Wikimedia FoundationLead sectionQuote, Lead section
This question was posed in 1930 by Gottfried Köthe (1905-1989).
A counterexample to Kothe's conjecture and a question of Rowen (arXiv preprint)
Abstract
We construct a counterexample to Kothe's conjecture and use this to also answer a question due to Rowen.
Submission date line
[Submitted on 7 Sep 2026]
View the SourceKöthe conjecture (Wikipedia)
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