Math 2.0

Nonassociative rings and algebras Basic notions of nonassociative rings and algebras Theorem

Shirshov–Cohn theorem

Builds on Jordan algebraSpecial Jordan algebra

Statement

Let KK be a field of characteristic not 22, and let JJ be a Jordan algebra over KK. Let x,y∈Jx, y \in J. Then the subalgebra of JJ generated by xx and yy is special. That is, it is isomorphic to a subalgebra of an algebra B+B^+ for some associative algebra BB over KK. Here B+B^+ is BB with the product a∘b=12(ab+ba)a \circ b = \frac{1}{2}(ab + ba).

Remarks

The theorem says that any Jordan algebra generated by two elements is special. It does not extend to three generators: the free Jordan algebra on three generators is not special. The core of the proof is that the free Jordan algebra on two generators is special. As a consequence, any identity in two variables that holds in all special Jordan algebras holds in all Jordan algebras.

Going deeper

For a student

The theorem is a two-variable analogue of Artin's theorem for alternative algebras. It lets one prove identities in two variables for all Jordan algebras by checking them in associative algebras. Macdonald's theorem extends this to identities in three variables that are of degree at most one in the third variable.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Jordan_algebra
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Kindstatement, theorem
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MSC17C05, 17C10
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AboutJordan algebra
UsesJordan algebra, Special Jordan algebra
Changed3 November 2026

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