On bases that are closed under multiplication

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.