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| Paper IPM / M / 7896 |
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| Abstract: | |
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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.
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