Permuting subalgebras of full matrix algebras (Q1057350)

Let R be a full matrix ring over a division ring D which is finite dimensional over its centre Z. Suppose U is a subalgebra of R satisfying (i) U.Z\(\leq U\) and (ii) \(U+U^ r\) is multiplicatively closed for all units r of R. Then, with the exception of \(R\cong M_ 2(GF(2))\), one of the following holds: (i) U is (0) or R (the trivial cases); (ii) U is a left (right) ideal of R; (iii) U is the sum of Z and a left (right) ideal of R; (iv) U is the sum of D and a maximal left (right) ideal of R. The proof is by induction on n, the degree of the matrices, but for \(1\leq n\leq 4\) a direct proof by means of elementary matrix calculations is given.