Math 2.0

Associative rings and algebras Polynomial identity rings Theorem

Kaplansky's theorem on PI rings

Builds on Primitive ringPolynomial identity ringCentral simple algebraCenter of a ringJacobson density theoremAmitsur–Levitzki theoremDivision ring

Statement

Let RR be a primitive ring (right or left). Assume that RR satisfies a monic polynomial identity of degree dd. Let ZZ be the center of RR. Then ZZ is a field, RR is a central simple algebra over ZZ, and dim⁡ZR=m2withm≤d/2.\dim_Z R = m^2 \quad \text{with} \quad m \le d/2.

Remarks

By the Jacobson density theorem, RR is a dense ring of linear maps of a vector space VV over a division ring DD. If dim⁡DV\dim_D V were large, a subring of RR would map onto a ring of matrices Ms(D)M_s(D) with ss large. Such a matrix ring fails every monic identity of degree less than 2s2s, so dim⁡DV\dim_D V is finite and R≅Ms(D)R \cong M_s(D). Then DD is a PI division ring, and a maximal subfield argument shows that DD is finite-dimensional over its centre. The bound m≤d/2m \le d/2 comes from comparing with matrix rings over a splitting field, using part 2 of the Amitsur–Levitzki theorem.

Example: R=Mn(F)R = M_n(F) for a field FF is primitive, satisfies the standard identity s2ns_{2n} of degree d=2nd = 2n, and has dim⁡FR=n2\dim_F R = n^2 with n=d/2n = d/2. So the bound is sharp.

The primitive hypothesis is needed: the polynomial ring ℤ[x]\Z[x] satisfies the identity x1x2−x2x1x_1 x_2 - x_2 x_1 of degree 22, but its centre ℤ[x]\Z[x] is not a field.

Going deeper

For a student

The theorem is the first structure theorem of PI theory. Because every semiprimitive ring is a subdirect product of primitive rings, it lets one embed semiprimitive PI rings into products of central simple algebras of bounded degree. It extends to prime PI rings through Posner's theorem, and it is one of the tools behind the positive answer to the Kurosh problem for PI algebras.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

Show the path as a graph
Center of a ring, concept, no evidence levelCenter of a ringconceptCentral simple algebra, concept, no evidence levelCentral simple algebraconceptCommutative ring, concept, no evidence levelCommutative ringconceptDivision ring, concept, no evidence levelDivision ringconceptEndomorphism ring, concept, no evidence levelEndomorphism ringconceptExterior algebra, concept, no evidence levelExterior algebraconceptFree algebra, concept, no evidence levelFree algebraconceptGroup, concept, no evidence levelGroupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionModule over a ring, definition, no evidence levelModule over a ringdefinitionPolynomial identity ring, concept, no evidence levelPolynomial identity r…conceptPrimitive ring, concept, no evidence levelPrimitive ringconceptSign of a permutation, concept, no evidence levelSign of a permutationconceptSimple module, concept, no evidence levelSimple moduleconceptSquare matrix, concept, no evidence levelSquare matrixconceptSymmetric group, concept, no evidence levelSymmetric groupconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionSchur's lemma, statement, no evidence levelSchur's lemmastatementLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionEndomorphism ring of a simple module is a division ring, statement, no evidence levelEndomorphism ring of …statementDivision ring, definition, no evidence levelDivision ringdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionJacobson density theorem, statement, no evidence levelJacobson density theo…statementCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCayley–Hamilton theorem, statement, no evidence levelCayley–Hamilton theor…statementAmitsur–Levitzki theorem, statement, no evidence levelAmitsur–Levitzki theo…statementKaplansky's theorem on PI rings, statement, no evidence levelKaplansky's theorem o…statement
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. Center of a ring concept
  2. Central simple algebra concept
  3. Commutative ring concept
  4. Division ring concept
  5. Endomorphism ring concept
  6. Exterior algebra concept
  7. Free algebra concept
  8. Group concept
  9. Identity matrix object
  10. Matrix definition
  11. Module over a ring definition
  12. Polynomial identity ring concept
  13. Primitive ring concept
  14. Sign of a permutation concept
  15. Simple module concept
  16. Square matrix concept
  17. Symmetric group concept
  18. Vector space definition
  19. Ring definition
  20. Square matrix definition
  21. Schur's lemma statement
  22. Linear map definition
  23. Unit of a ring definition
  24. Endomorphism definition
  25. Multilinear map definition
  26. Endomorphism ring of a simple module is a division ring statement
  27. Division ring definition
  28. Polynomial of an endomorphism definition
  29. Alternating multilinear map definition
  30. Determinant definition
  31. Jacobson density theorem statement
  32. Characteristic polynomial definition
  33. Cayley–Hamilton theorem statement
  34. Amitsur–Levitzki theorem statement
  35. Kaplansky's theorem on PI rings statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_identity_ring
About this page
Tag8NE9Q44FEG
URIm2://8NE9Q44FEG
Kindstatement, theorem
WritingW0 stub
MSC16R20, 16D60
StatusThis copy of the site has no build data. Run m2 status 8NE9Q44FEG.
AboutPolynomial identity ring, Primitive ring
UsesPrimitive ring, Polynomial identity ring, Central simple algebra, Center of a ring, Jacobson density theorem, Amitsur–Levitzki theorem, Division ring

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