Jordan ring

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Definition

Symbol-free definition

A Jordan ring is a non-associative ring (i.e., a not necessarily associative ring) whose multiplication gives a Jordan magma.

Definition with symbols

A Jordan ring is a set A equipped with binary operations + and \cdot, a constant 0, and a unary operation -, such that:

  1. (A,+,0,-) is an abelian group.
  2. Distributivity laws: For all a,b,c \in A:
    • \! a \cdot (b + c) = a \cdot b + a \cdot c
    • \! (a + b) \cdot c = a \cdot c + b \cdot c.
  3. Commutativity of \cdot: For all a,b \in A, a \cdot b = b \cdot a.
  4. The Jordan identity: For all a,b \in A, we have:

\! (a \cdot b) \cdot (a \cdot a) = a \cdot (b \cdot (a \cdot a)).