Alternative ring

Definition

An alternative ring is a non-associative ring $R$ (i.e., a not necessarily associative ring) that, under its multiplication $*$, satisfies one of the following equivalent conditions:

1. $(R,*)$ is an alternative magma, i.e., it satisfies the identities $x*(x*y)=(x*x)*y$ and $x*(y*y)=(x*y)*y$ for all $x,y\in R$.
2. $(R,*)$ is both a left-alternative magma and a flexible magma, i.e., it satisfies the identities $x*(x*y)=(x*x)*y$ and $x*(y*x)=(x*y)*x$ for all $x,y\in R$.
3. $(R,*)$ is both a right-alternative magma and a flexible magma, i.e., it satisfies the identities $x*(y*y)=(x*y)*y$ and $x*(y*x)=(x*y)*x$ for all $x,y\in R$.
4. The subring of $R$ generated by any two elements of $R$ is an associative ring.

Equivalence of definitions

• The equivalence of definitions (1) -- (3) follows by looking at the associator on a non-associative ring and proving that any of the conditions is equivalent to that associator being an alternating function of its inputs. Further information: equivalence of definitions of alternative ring
• The equivalence with definition (4) is Artin's theorem on alternative rings.