On bases that are closed under multiplication

TL;DR

This paper extends a known result about bases closed under multiplication from real numbers to a broader class of algebras, including matrix algebras, by adapting existing proofs.

Contribution

It generalizes the non-existence of multiplicatively closed bases from real fields to a wider class of algebraic structures, including matrix algebras.

Findings

No basis of the real numbers over any proper subfield is closed under multiplication.

The proof for real numbers can be adapted to larger classes of algebras.

Includes matrix algebras as a case where such bases do not exist.

Abstract

It is well known that there is no basis of the field for real numbers regarded as a vector space over any proper subfield that is closed under multiplication. Mabry has extended this result to bases of arbitrary proper field extensions. The aim of this short communication is to notice that the proof of the result concerning the reals may be adjusted to a larger class of algebras (including full matrix algebras); thereby we subsume Mabry's result.