---
tag: RT2D9JEVRR
kind: conjecture
name: Köthe conjecture
msc: [16N40]
uses: [0894J980RK, 5N2CJZ47D4, NZ1G9WRWFB, 3G2JQEYBEK]
about: [3G2JQEYBEK]
sources:
  - url: https://en.wikipedia.org/wiki/K%C3%B6the_conjecture
same_as:
  wikidata: Q6454025
  wikipedia: Köthe conjecture
history:
  posed_by: Gottfried Köthe
  first_stated: "1930"
---

## Statement

Let $R$ be a [[ring|0894J980RK]].
A one-sided ideal of $R$ is *nil* if each of its elements is nilpotent.
The conjecture states that if $R$ has no nonzero [[nil ideal|NZ1G9WRWFB]] (two-sided), then $R$ has no nonzero nil [[left ideal|5N2CJZ47D4]].

## 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|3G2JQEYBEK]].
Krempa showed that it is equivalent to the statement that for every nil ring $R$ the polynomial ring $R[x]$ is equal to its Jacobson radical.
It holds for left Noetherian rings, where every nil one-sided ideal is nilpotent by [[Levitzki's theorem|QHD4KJYFSA]].
Smoktunowicz constructed a nil ring $R$ such that $R[x]$ is not nil, which rules out a stronger form that had been suggested.

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
In a noncommutative ring there are several ways to collect nilpotent elements: in two-sided ideals or in one-sided ideals.
The conjecture says that one-sided ideals of nilpotent elements can only exist if a two-sided one does.
It is one of the oldest open problems of ring theory.