Math 2.0

Associative rings and algebras Chain, growth and finiteness conditions Theorem

Lenagan's theorem

Builds on RingTwo-sided idealNil idealNilpotent idealUpper nilradicalArtinian moduleModule over a ringKrull dimension of a noncommutative ringLevitzki's theorem

Statement

Let RR be a ring with identity. For a right RR-module MM, the Krull dimension Kdim⁡M\operatorname{Kdim} M (in the sense of Gabriel and Rentschler) is defined by transfinite recursion on ordinals. One sets Kdim⁡0=−1\operatorname{Kdim} 0 = -1. Let α≥0\alpha \ge 0 be an ordinal, and consider descending chains M0⊇M1⊇⋯M_0 \supseteq M_1 \supseteq \cdots of submodules of MM. Then Kdim⁡M≤α\operatorname{Kdim} M \le \alpha means: in every such chain, all but finitely many factors Mi/Mi+1M_i / M_{i+1} have Krull dimension less than α\alpha. If such ordinals exist, Kdim⁡M\operatorname{Kdim} M is the least of them; otherwise MM has no Krull dimension. The ring RR has right Krull dimension if Kdim⁡RR≤α\operatorname{Kdim} R_R \le \alpha for some ordinal α\alpha.

Assume that RR has right Krull dimension. Then the upper nilradical of RR is a nilpotent ideal. Equivalently, every nil ideal of RR, that is, every two-sided ideal whose elements are all nilpotent, is nilpotent.

Remarks

The nonzero modules of Krull dimension 00 are exactly the nonzero Artinian modules. Every right Noetherian ring has right Krull dimension. So the theorem contains the two-sided case of Levitzki's theorem: in a right Noetherian ring every nil ideal is nilpotent. It also covers right Artinian rings, where the nil ideals lie in the Jacobson radical, which is nilpotent.

An outline of the proof: one first shows that the prime radical NN of RR is nilpotent. The quotient R/NR/N is semiprime and still has right Krull dimension, which makes it a right Goldie ring. A semiprime right Goldie ring has no nonzero nil ideals, so every nil ideal of RR lies in NN.

The hypothesis cannot be dropped. Let kk be a field and R=k[x1,x2,… ]/(x1,x22,x33,… )R = k[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots), with xˉi\bar x_i the image of xix_i. The ideal II generated by all xˉi\bar x_i is nil, since each element of II lies in an ideal generated by finitely many nilpotent elements of a commutative ring. It is not nilpotent, since xˉn n−1≠0\bar x_n^{\,n-1} \ne 0 lies in In−1I^{n-1} for every nn. This ring has no Krull dimension.

Going deeper

For a student

Krull dimension of a module measures how far the module is from being Artinian, by a deviation of its lattice of submodules. Rings with right Krull dimension include all right Noetherian and all right Artinian rings, and many rings that are neither. The theorem extends the classical facts that nil ideals are nilpotent in right Artinian rings and, by Levitzki's theorem, in right Noetherian rings. It is a typical use of Krull dimension: a statement proved for Noetherian rings by an ascending chain argument is carried over by induction on the deviation. The commutative example k[x1,x2,… ]/(x1,x22,x33,… )k[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots) shows that some finiteness condition is needed even without noncommutativity.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

Show the path as a graph
Annihilator, concept, no evidence levelAnnihilatorconceptArtinian module, concept, no evidence levelArtinian moduleconceptArtinian ring, concept, no evidence levelArtinian ringconceptAscending chain condition, definition, no evidence levelAscending chain condi…definitionComposition series, concept, no evidence levelComposition seriesconceptDivision ring, concept, no evidence levelDivision ringconceptGroup, concept, no evidence levelGroupconceptJacobson radical, concept, no evidence levelJacobson radicalconceptKrull dimension of a noncommutative ring, concept, no evidence levelKrull dimension of a …conceptLeft ideal, concept, no evidence levelLeft idealconceptMatrix ring, object, no evidence levelMatrix ringobjectModule over a ring, concept, no evidence levelModule over a ringconceptNil ideal, concept, no evidence levelNil idealconceptNilpotent element, concept, no evidence levelNilpotent elementconceptNilpotent ideal, concept, no evidence levelNilpotent idealconceptNoetherian module, concept, no evidence levelNoetherian moduleconceptNoetherian ring, concept, no evidence levelNoetherian ringconceptRing, concept, no evidence levelRingconceptSemiprimary ring, concept, no evidence levelSemiprimary ringconceptSemisimple module, concept, no evidence levelSemisimple moduleconceptSemisimple ring, concept, no evidence levelSemisimple ringconceptTwo-sided ideal, concept, no evidence levelTwo-sided idealconceptUpper nilradical, concept, no evidence levelUpper nilradicalconceptRing, definition, no evidence levelRingdefinitionArtin–Wedderburn theorem, statement, no evidence levelArtin–Wedderburn theo…statementCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionIdeal, definition, no evidence levelIdealdefinitionArtinian ring, definition, no evidence levelArtinian ringdefinitionIdeal generated by a subset, definition, no evidence levelIdeal generated by a …definitionPrime ideal, definition, no evidence levelPrime idealdefinitionMaximal ideal, definition, no evidence levelMaximal idealdefinitionNoetherian ring, definition, no evidence levelNoetherian ringdefinitionJacobson radical of an Artinian ring is nilpotent, statement, no evidence levelJacobson radical of a…statementAkizuki–Hopkins–Levitzki theorem, statement, no evidence levelAkizuki–Hopkins–Levit…statementHopkins–Levitzki theorem, statement, no evidence levelHopkins–Levitzki theo…statementLevitzki's theorem, statement, no evidence levelLevitzki's theoremstatementLenagan's theorem, statement, no evidence levelLenagan's theoremstatement
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. Annihilator concept
  2. Artinian module concept
  3. Artinian ring concept
  4. Ascending chain condition definition
  5. Composition series concept
  6. Division ring concept
  7. Group concept
  8. Jacobson radical concept
  9. Krull dimension of a noncommutative ring concept
  10. Left ideal concept
  11. Matrix ring object
  12. Module over a ring concept
  13. Nil ideal concept
  14. Nilpotent element concept
  15. Nilpotent ideal concept
  16. Noetherian module concept
  17. Noetherian ring concept
  18. Ring concept
  19. Semiprimary ring concept
  20. Semisimple module concept
  21. Semisimple ring concept
  22. Two-sided ideal concept
  23. Upper nilradical concept
  24. Ring definition
  25. Artin–Wedderburn theorem statement
  26. Commutative ring definition
  27. Unit of a ring definition
  28. Ideal definition
  29. Artinian ring definition
  30. Ideal generated by a subset definition
  31. Prime ideal definition
  32. Maximal ideal definition
  33. Noetherian ring definition
  34. Jacobson radical of an Artinian ring is nilpotent statement
  35. Akizuki–Hopkins–Levitzki theorem statement
  36. Hopkins–Levitzki theorem statement
  37. Levitzki's theorem statement
  38. Lenagan's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nil_ideal
About this page
TagQSK2J5HEER
URIm2://QSK2J5HEER
Kindstatement, theorem
WritingW0 stub
MSC16N40, 16P60, 16P40
StatusThis copy of the site has no build data. Run m2 status QSK2J5HEER.
AboutNil ideal, Nilpotent ideal, Upper nilradical
UsesRing, Two-sided ideal, Nil ideal, Nilpotent ideal, Upper nilradical, Artinian module, Module over a ring, Krull dimension of a noncommutative ring, Levitzki's theorem

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