Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetSn(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F*, thenT is an additive injective operator preserving rank-additivity onSn(F) if and only if there exists an invertible matrixU∈Mn(F) and an injective field homomorphism ϕ ofF to itself such thatT(X)=cUXϕUT, ∀X=(xij)∈Sn(F) wherec∈F*,Xϕ=(ϕ(xij)). As applications, we determine the additive operators preserving minus-order onSn(F) over the fieldF.