This paper deals with the
problem of proving that a simple algebra (finite dimensional) has an identity element.
The main result is contained in the following theorem. Let A be a simple algebra
(char. ≠2) in which (x,x,x) = 0 and x3⋅ x = x2⋅ x2. If M is a subset of A such
that (A,M,A) = 0 and (M,A,A) ∪ (M,A) ∪ (A,A,M) ⊆ M, then M = 0
or there is an identity element in A. This result is then used to prove the
three following corollaries (char. ≠2): (1) A simple power associative algebra
with all commutators in the nucleus has an identity; (2) A simple power
associative algebra with all associators in the middle center has an identity; (3) A
simple antiflexible algebra in which (x,x,x) = 0 and A+ is not nil has an
identity.
