Burnsideʹs theorem for matrix rings over division rings Original Research Article

A version of Burnsideʹs theorem states that if F is an arbitrary field and image is an irreducible (or, equivalently, transitive) subalgebra containing a rank-one matrix, then image. The present paper shows that if F is replaced by a division ring D, then every transitive left subalgebra of Mn(D) containing a rank-one matrix is equal to Mn(D). (Here, by a left algebra we mean a ring which is also a left D-module.) Counterexamples are given in case image is irreducible but not transitive. Moreover, it is shown that irreducible left algebras of quaternionic matrices contain rank-one idempotents and their structures are classified.