On cyclic left modules and arbitrary bimodules over the octonions

TL;DR

This paper completes the classification of cyclic left modules over the octonions and provides a comprehensive description of bimodules, addressing previously unresolved questions in the structure theory of octonionic modules.

Contribution

It offers a complete characterization of cyclic left modules and an explicit description of bimodules over the octonions, filling a gap in existing literature.

Findings

Complete classification of cyclic left $O$-modules

Explicit structure description of bimodules over $O$

Addresses open questions from previous work

Abstract

Qinghai Huo, Yong Li, Guangbin Ren described the structure of left $\mathbb{O}$-modules in great detail in arXiv:1911.08282. However, they left open a question on cyclic left $\mathbb{O}$-modules. This note intends to close this gap and provide a complete description of bimodules.