Additive preservers of non-zero decomposable tensors

Let T be an additive mapping from a tensor product of vector spaces over a field into itself. We describe T for the following two cases: (i) T is surjective and sends non-zero decomposable elements to non-zero decomposable elements, and (ii) T(A) is a non-zero decomposable element if and only if A is a non-zero decomposable element.