On irreducible and transitive subalgebras in matrix algebras

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Let R = M n (D) be the n × n matrix ring over a division ring D and let A be a subring of R with r as the smallest non-zero rank present in A. It is shown that A is a transitive subring of R if and only if r divides n and after a similarity A = M n/r (Δ), where Δ is a transitive division subring of M r (D). Let A be an F-subalgebra of R with 1 R ∈ A, where F is the centre of R, and let C R (A) denote the centralizer of A in R. Using the above result, we describe when the centralizer C R (A) is transitive or irreducible in R. Further assume that D is finite dimensional over F. It is shown that C R (A) is a transitive subalgebra of R if and only if the tensor product is a division algebra. In this case, dim(A F ) is the smallest non-zero rank present in C R (A).