Math 2.0

Associative rings and algebras Conditions on elements Conjecture

Jacobson's conjecture

Builds on RingJacobson radicalNoetherian ring

Statement

Let RR be a ring with unit that is Noetherian on both sides, that is, left Noetherian and right Noetherian. Let JJ be the Jacobson radical of RR. Then ⋂n≥1Jn=0.\bigcap_{n \ge 1} J^n = 0.

Remarks

The conjecture is open in general. For commutative Noetherian rings it holds, by the Krull intersection theorem. The one-sided version, for rings that are only right Noetherian, is false; I. N. Herstein gave a counterexample. The conjecture is known for several classes of two-sided Noetherian rings, among them the fully bounded Noetherian rings.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Ascending chain condition, definition, no evidence levelAscending chain condi…definitionGroup, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionIdeal, definition, no evidence levelIdealdefinitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionNoetherian ring, definition, no evidence levelNoetherian ringdefinitionJacobson radical, definition, no evidence levelJacobson radicaldefinitionJacobson's conjecture, conjecture, no evidence levelJacobson's 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. Ascending chain condition definition
  2. Group concept
  3. Ring definition
  4. Commutative ring definition
  5. Unit of a ring definition
  6. Ideal definition
  7. Ideal generated by a subset definition
  8. Maximal ideal definition
  9. Noetherian ring definition
  10. Jacobson radical definition
  11. Jacobson's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Jacobson%27s_conjecture
About this page
TagBFP0ZST7V3
URIm2://BFP0ZST7V3
Kindconjecture
WritingW0 stub
MSC16P40, 16N20
StatusThis copy of the site has no build data. Run m2 status BFP0ZST7V3.
AboutJacobson radical, Noetherian ring
UsesRing, Jacobson radical, Noetherian ring
Same as
Changed2 November 2026

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