Additive preservers of rank on alternate matrix spaces over fields and applications

Suppose F is any field and n is an integer with n ⩾ 4. Let Kn(F) be the set of all n × n alternate matrices over F, and let (Kn(F), +, ·) be the non-associative ring formed by Kn(F) under the usual addition '+' and the multiplication '·' defined by X · Y = XYX for all X, Y ∈ Kn(F). A pair of n × n matrices (A, B) is said to be rank-additive if rank(A + B) = rank A + rank B, and rank-subtractive if rank(A − B) = rank A − rank B. We say that an operator ϕ :Kn(F) → Kn(F) is additive if ϕ(X + Y) = ϕ(X) + ϕ(Y) for any X, Y ∈ Kn(F), a preserver of rank-additivity (respectively, rank-subtractivity) on Kn(F) if it preserves the set of all rank-additive (respectively, rank-subtractive) pairs, a preserver of rank on Kn(F) if rank ϕ(X) = rank X for every X ∈ Kn(F), and a ring endomorphism of (Kn(F), +, ·) if it is additive and satisfies ϕ(X ·Y) = ϕ(X) · ϕ(Y) for any X, Y ∈ Kn(F). We determine the general form of all additive preservers of rank (respectively, rank-additivity and rank-subtractivity) on Kn(F) and characterize all ring endomorphisms of (Kn(F), +, ·).