Linear operators preserving idempotence on matrices spaces over skew-fields

摘要

LET R and R_1 be skew-fields with centres F and F_1, where F is contained in F_1 and | F | >2. By M_n (R) and I_n (R) we denote the F-space of all n X n matrices over R and the set of all idempotent matrices in M_n (R), respectively. If a linear map Lfrom M_n (R) to M_m (R_r) satisfies L (I_n (R)) is contained in I_m (R_1) we call L an idempotence preserver (all such maps will be denoted by L_(n,m) (R, R_1)). To determine the forms of idempotence preservers is one important content of the linear preserver problems (see refs.) and is the purpose of this note. Suppose that [ R, R ] is the F- subspace of R generated by all finite sums of elements with the form of ab-ba, and R = R/[R, R].