Math 2.0

Associative rings and algebras Radicals Conjecture

Köthe conjecture

Builds on RingLeft idealNil idealUpper nilradical

Statement

Let RR be a ring. A one-sided ideal of RR is nil if each of its elements is nilpotent. The conjecture states that if RR has no nonzero nil ideal (two-sided), then RR has no nonzero nil left ideal.

Remarks

The conjecture is open. Equivalent forms: in every ring the sum of two nil left ideals is nil; in every ring every nil left ideal is contained in the upper nilradical. Krempa showed that it is equivalent to the statement that for every nil ring RR the polynomial ring R[x]R[x] is equal to its Jacobson radical. It holds for left Noetherian rings, where every nil one-sided ideal is nilpotent by Levitzki's theorem. Smoktunowicz constructed a nil ring RR such that R[x]R[x] is not nil, which rules out a stronger form that had been suggested.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptLeft ideal, concept, no evidence levelLeft idealconceptNil ideal, concept, no evidence levelNil idealconceptUpper nilradical, concept, no evidence levelUpper nilradicalconceptRing, definition, no evidence levelRingdefinitionKöthe conjecture, conjecture, no evidence levelKöthe conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Left ideal concept
  3. Nil ideal concept
  4. Upper nilradical concept
  5. Ring definition
  6. Köthe conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/K%C3%B6the_conjecture
About this page
TagRT2D9JEVRR
URIm2://RT2D9JEVRR
Kindconjecture
WritingW0 stub
MSC16N40
StatusThis copy of the site has no build data. Run m2 status RT2D9JEVRR.
AboutUpper nilradical
UsesRing, Left ideal, Nil ideal, Upper nilradical
Same as
Changed2 November 2026

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