Large annihilators in Cayley-Dickson algebras

2005/11/28 by Daniel K. Biss, Biss, Daniel K., Daniel Dugger +3
Engineering · Mathematics · #Advanced Materials and Mechanics #Algebraic Topology (math.AT) #Algebraic and Geometric Analysis #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.math/0511691

openalex publication_date 2005/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Cayley-Dickson algebras are an infinite sequence of non-associative algebras starting with the reals, complexes, quaternions, and octonions. We study the zero-divisors in the higher Cayley-Dickson algebras. In particular, we show that the annihilator of any element in the 2ⁿ-dimensional Cayley-Dickson algebra has dimension at most 2ⁿ-4n+4. Moroever, every multiple of four between 0 and this upper bound actually occurs as the dimension of some annihilator (a theorem of Moreno says that only multiples of four can occur). We completely describe all the zero-divisors whose annihilator has dimension 2ⁿ-4n+4.

Related

• The zero divisors of the Cayley-Dickson algebras over the real numbers
• Constructing zero divisors in the higher dimensional Cayley-Dickson algebras
• Structure of the Cayley-Dickson algebras
• About some properties of algebras obtained by the Cayley-Dickson process
• Some remarks about the levels and sublevels of algebras obtained by the\n Cayley-Dickson process