On F-algebras of algebraic matrices over a subfield F of the center of a division ring

Let D be a division ring and F a subfield of its center. We prove a Wedderburn-Artin type theorem for irreducible F-algebras of F-algebraic matrices in Mn(D). We then use our result to show that, up to a similarity, Mn(F) is the only irreducible F-algebra of triangularizable matrices in Mn(D) with inner eigenvalues in F provided that such an F-algebra exists. We use this result to prove a block triangularization theorem, which is a well-known result for algebras of matrices over algebraically closed fields, for F-algebras of triangularizable matrices in Mn(D) with inner eigenvalues in the subfield F of the center of D. We use our main results to prove the counterparts of some classical and new triangularization results over a general division ring. Also, we generalize a well-known theorem of W. Burnside to irreducible F-algebras of matrices in Mn(K) with traces in the subfield F of the field K.