Linear/additive preservers of ranks 2 and 4 on alternate matrix spaces over fields

Let K-n (F) be the linear space of all n x n alternate matrices over a field F. An operator f : K-n (F) --> K-n (F) is said to be additive if f (A + B) = f(A) + f (B) for any A, B is an element of K-n (F), linear if f is additive and f (aA) = af (A) for every a is an element of F and A is an element of K-n (F), and a preserver of ranks 2 and 4 on K-n (F) if rank f (X) = rankX for every X is an element of K-n (F) with rankX = 2 or 4. When n greater than or equal to 4, we characterize all linear (respectively, additive) preservers of ranks 2 and 4 on K-n (F) over any field (respectively, any field that is not isomorphic to a proper subfield of itself). Furthermore, it is also shown that the condition "F is not isomorphic to a proper subfield of itself " is necessary for the obtained conclusion on additive preservers of ranks 2 and 4 on K-n(F). (C) 2004 Elsevier Inc. All rights reserved.