# Cayley-Dickson Process and Centrally Essential Rings

## TL;DR

This paper investigates the structure of rings formed via the Cayley-Dickson process, focusing on their centers and associative centers, and provides conditions for these rings to be centrally essential, including examples of non-associative cases.

## Contribution

It characterizes the associative and center structures of rings from the Cayley-Dickson process and establishes conditions for these rings to be centrally essential, with applications to quaternion and octonion rings.

## Key findings

- Conditions for rings to be N-essential or centrally essential.
- Application to quaternion and octonion rings.
- Construction of a non-associative centrally essential ring.

## Abstract

We describe associative center $N(R)$ and the center $Z(R)$ of the ring $R$ obtained by applying the generalized Cayley-Dickson construction and we find conditions under which the ring $R$ is $N$-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a non-associative centrally essential ring. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.