Artin's generalized theorem on alternative rings

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Statement

Suppose R is an alternative ring and x,y,z are (possibly equal, possibly distinct) elements of R. Then the following are equivalent, where a denotes the associator:

  • a(x,y,z) = 0
  • a(x,y,z) = a(x,z,y) = a(y,x,z) = a(y,z,x) = a(z,x,y) = a(z,y,x) = 0
  • The subring of R generated by x,y,z is associative.

Facts used

  1. Subset version of Artin's generalized theorem on alternative rings

Related facts

Applications