Burnside's theorem for matrix rings over division rings (Q1826831)

The setting for this paper is a left subalgebra \({\mathcal A}\) of the matrix algebra \(M_n(D)\) over a general division ring \(D\) (to wit, \({\mathcal A}\) is a subring of \(M_n(D)\) which is a left \(D\)-module, too). A version of Burnside's theorem is here presented by showing that every transitive (and hence irreducible) left subalgebra of \(M_n(D)\) containing a rank-one matrix is equal to \(M_n(D)\). The last condition compensates for the lack of the requirement that \(D\) be an algebraically closed field. Counterexamples are also given in case \(D\) is the ring \({\mathbf H}\) of real quaternions and \(n\) is any even integer, where \({\mathcal A}\) is irreducible but not transitive, contrary to the commutative case.