  In this paper we study the algebra monomorphisms from A_m =R^(2^m) into
A_n=R^(2^n) for 0<m<n where the A_k 's are the Cayley- Dickson algebras over
the real numbers. We show that for m>2 there are many different types of
monomorphisms and we describe them in terms of the zero divisors in A_n.
