Jordan ring

Symbol-free definition
A Jordan ring is a non-associative ring (i.e., a not necessarily associative ring) whose multiplication gives a defining ingredient::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 $$a,b,c \in A$$:
 * 3) * $$\! a \cdot (b + c) = a \cdot b + a \cdot c$$
 * 4) * $$\! (a + b) \cdot c = a \cdot c + b \cdot c$$.
 * 5) Commutativity of $$\cdot$$: For all $$a,b \in A$$, $$a \cdot b = b \cdot a$$.
 * 6) The Jordan identity: For all $$a,b \in A$$, we have:

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