Jordan implies power-associative

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This article gives the statement and possibly, proof, of an implication relation between two non-associative ring properties. That is, it states that every non-associative ring satisfying the first non-associative ring property (i.e., Jordan ring) must also satisfy the second non-associative ring property (i.e., power-associative ring)
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Statement

Any Jordan ring is a power-associative ring. In other words, if R is a commutative possibly non-associative ring satisfying the following

x * (y * (x * x)) = (x * y) * (x * x) \ \forall \ x,y \in R

Related facts

The analogous statement is not necessarily true for Jordan magmas, i.e., a Jordan magma need not be a power-associative magma. However, it is true that Jordan implies powers up to the fifth are well-defined even in the magma context.

Facts used

  1. Jordan implies powers up to the fifth are well-defined: Actually, we only need the fourth power to be well-defined.
  2. Equivalence of definitions of power-associative ring: This in particular implies that a non-associative ring is power-associative iff fourth powers are well-defined in the ring.

Proof

The proof combines facts (1) and (2).