GENERALIZATIONS OF THE CAYLEY-DICKSON DOUBLING PROCEDURE FOR ALGEBRAS OVER C
The Cayley-Dickson doubling procedure applies to any finite dimensional algebra and yields a new algebra whose dimension is twice that of the original algebra. Corresponding to this procedure is extended to any natural prime number p. We call p-ling of an algebra this extended procedure. The iterated application of a p-ling to any finite dimensional algebra over C generates an infinite sequence of algebras, each one having a dimension equal to p times that of the preceding algebra in the sequence. The p-ling of an algebra may use the conjugation defined on this algebra. We examine the properties of a particular p-ling based on a multiplication generalizing that of complex numbers. All algebras resulting from this p-ling without conjugation admit divisors of zero and a multiplicative pseudo-norm. For the algebra obtained from R through this p-ling without conjugation is expressible as the direct sum of a family made up of a straight line and planes. The divisors of zero of are the elements of the direct sum of one or the other of the proper subfamilies of which is canceled by the direct sum of its complement subfamily in
algebras of hypercomplex numbers, generalized doubling procedure, multiplicative pseudo-norm.