Algebraic extensions of some results by Yang

Abstract

By means of differential geometry arguments Yang showed that for every unital real division algebra $\mathcal{A}$ of dimension > 1 the square map $x\mapsto x^2$ is onto and, consequently, $\mathcal{A}$ contains a subalgebra isomorphic to $\mathbb{C}.$ The first assertion is extended algebraically for an algebra with only a non-zero flexible element. It persists if the unit is replaced by a non-zero central element. Next, always by algebraic approach, we give examples of both four-dimensional and eight-dimensional real division algebras $\mathcal{A}$ with left-unit e containing no two-dimensional subalgebras with the additional property $R_e^2=I_{\mathcal{A}}.$

KEYWORDS:

• Central (flexible) element (idempotent)
• left-unit
• real division algebra

Acknowledgement

The authors are very grateful to the Referee for his valuable comments which led to an improvement of the manuscript.