Until thirty years into this century the theory of rings developed mainly
as the theory of associative rings. However, even in the middle of the last
century there arose mathematical systems that satisfied all the axioms of a
ring except associativity. For example, the algebra of Cayley numbers, which
was constructed in 1845 by the English mathematician Arthur Cayley, is such
a system. An identity found by Degen in 1818, which represents the product
of two sums of eight squares again in the form of a sum of eight squares,
was rediscovered by means of this algebra. The algebra of Cayley numbers
is an eight-dimensional division algebra over the field of real numbers, which
satisfies the following weakened identities of associativity: (m)b = a(&),
(ab)b = a(bb).
Algebras satisfying these two identities were subsequently named alternative.
Such a name stems from the fact that in every algebra satisfying the
two indicated identities, the associator (x, y, 2) = (xy)z - x(yz) is an alternating
(skew-symmetric) function of its arguments. The theory of alternative
algebras attracted the serious attention of mathematicians after the discovery
of its deep connection with the theory of projective planes, which was
actively developed at the beginning of this century. In this regard, it was
discovered that alternative algebras are “sufficiently near” to associative
ones. The essence of this nearness is exhibited by the theorem of Artin, which
asserts that in every alternative algebra the subalgebra generated by any two
elements is associative.