Burnside's theorem in the setting of general fields

Abstract

We extend a well-known theorem of Burnside in the setting of general fields as follows: for a general field $F$ the matrix algebra $M_n(F)$ is the only algebra in $M_n(F)$ which is spanned by an irreducible semigroup of triangularizable matrices. In other words, for a semigroup of triangularizable matrices with entries from a general field irreducibility is equivalent to absolute irreducibility. As a consequence of our result we prove a stronger version of a theorem of Janez Bernik.