Math 2.0

Associative rings and algebras Polynomial identity rings Theorem

Amitsur–Levitzki theorem

Builds on Commutative ringSquare matrixSymmetric groupSign of a permutationPolynomial identity ringFree algebraExterior algebraCayley–Hamilton theorem

Statement

Let KK be a commutative ring with identity and let n≥1n \ge 1. For m≥1m \ge 1, the standard polynomial of degree mm is sm(x1,…,xm)=∑σ∈Smsgn⁡(σ) xσ(1)xσ(2)⋯xσ(m),s_m(x_1, \ldots, x_m) = \sum_{\sigma \in S_m} \sgn(\sigma) \, x_{\sigma(1)} x_{\sigma(2)} \cdots x_{\sigma(m)}, where SmS_m is the symmetric group and sgn⁡\sgn is the sign of a permutation.

  1. For all matrices A1,…,A2n∈Mn(K)A_1, \ldots, A_{2n} \in M_n(K), one has s2n(A1,…,A2n)=0s_{2n}(A_1, \ldots, A_{2n}) = 0.
  2. If K≠0K \ne 0, then Mn(K)M_n(K) satisfies no monic polynomial identity of degree less than 2n2n.

Remarks

Part 2 is the easy part. A monic identity of degree d<2nd < 2n can be replaced by a multilinear monic one of the same degree. One evaluates it on the first dd matrix units of the staircase e11,e12,e22,e23,…,en−1,n,enne_{11}, e_{12}, e_{22}, e_{23}, \ldots, e_{n-1,n}, e_{nn}. Only the product in the staircase order is nonzero, so the identity would have to have coefficient 00 on that monomial, and after renaming variables this contradicts monicity. For part 1 there are several proofs: the original combinatorial one, a short proof by Rosset that works in the exterior algebra, and proofs based on the Cayley–Hamilton theorem.

Example: for n=1n = 1, part 1 says ab−ba=0ab - ba = 0 for scalars a,ba, b, which is commutativity. For n=2n = 2, it says that the alternating sum of the 2424 products of four 2×22 \times 2 matrices in all orders is 00.

The degree 2n2n is the best possible by part 2: Mn(K)M_n(K) does not satisfy s2n−1s_{2n-1}.

Going deeper

For a student

The theorem says that Mn(K)M_n(K) is a polynomial identity ring with the standard identity s2ns_{2n}, and that 2n2n is the minimal degree of an identity. This fact is the base of the theory of PI rings, since by Kaplansky's and Posner's theorems many PI rings embed in matrix rings over commutative rings or fields. The minimal degree 2n2n of an identity is then a measure of the size of the matrices needed.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Commutative ring, concept, no evidence levelCommutative ringconceptExterior algebra, concept, no evidence levelExterior algebraconceptFree algebra, concept, no evidence levelFree algebraconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionPolynomial identity ring, concept, no evidence levelPolynomial identity r…conceptSign of a permutation, concept, no evidence levelSign of a permutationconceptSquare matrix, concept, no evidence levelSquare matrixconceptSymmetric group, concept, no evidence levelSymmetric groupconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionPolynomial of an endomorphism, definition, no evidence levelPolynomial of an endo…definitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionCharacteristic polynomial, definition, no evidence levelCharacteristic polyno…definitionCayley–Hamilton theorem, statement, no evidence levelCayley–Hamilton theor…statementAmitsur–Levitzki theorem, statement, no evidence levelAmitsur–Levitzki theo…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. Commutative ring concept
  2. Exterior algebra concept
  3. Free algebra concept
  4. Identity matrix object
  5. Matrix definition
  6. Polynomial identity ring concept
  7. Sign of a permutation concept
  8. Square matrix concept
  9. Symmetric group concept
  10. Vector space definition
  11. Square matrix definition
  12. Linear map definition
  13. Endomorphism definition
  14. Multilinear map definition
  15. Polynomial of an endomorphism definition
  16. Alternating multilinear map definition
  17. Determinant definition
  18. Characteristic polynomial definition
  19. Cayley–Hamilton theorem statement
  20. Amitsur–Levitzki theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Amitsur–Levitzki_theorem
About this page
TagYDH60KPTWT
URIm2://YDH60KPTWT
Kindstatement, theorem
WritingW0 stub
MSC16R10, 15A24
StatusThis copy of the site has no build data. Run m2 status YDH60KPTWT.
AboutPolynomial identity ring, Square matrix
UsesCommutative ring, Square matrix, Symmetric group, Sign of a permutation, Polynomial identity ring, Free algebra, Exterior algebra, Cayley–Hamilton theorem
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Changed2 November 2026

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