elements is therefore closed under multiplication and forms a Moufang loop. This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra. Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center. The structure theory of alternative rings is presented in. Applications The projective plane over any alternative division ring is a Moufang plane. The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows (page 162) the relation for an algebra A with unit element e and an involutive anti-automorphism such that a + a* and aa* are on the line spanned by e for all a in A. Use the notation n(a) = aa*. Then if n is a non-singular mapping into the field of A, and A is alternative, then (A,n) is a composition algebra. See also Algebra over a field Maltsev algebra Zorn ring References External links Non-associative algebras