Diagonability of idempotent matrices over noncommutative rings

Let R be an arbitrary ring. In this paper, the following statements are proved: (a) Each idempotent matrix over R can be diagonalized if and only if each idempotent matrix over R has a characteristic vector. (b) An idempotent matrix over R can be diagonalized under a similarity transformation if and only if it is equivalent to a di-agonal matrix. (a) and (b) generalize Foster’s and Steger’s theorems to arbitrary rings. We give some new results about 0-similarity of idempotent matrices over R. Ó 199