Jordan ring

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:

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