Structure theorems for rings under certain coactions of a Hopf algebra

Let {D-1, ... , D-n} be a system of derivations of a k-algebra A, k a field of characteristic p > 0, defined by a coaction delta of the Hopf algebra H-c = k[X-1, ... , X-n]/(X-1(p), ... , X-n(p)), c is an element of {0, 1}, the Lie Hopf algebra of the additive group and the multiplicative group on A, respectively. If there exist x(1), ... , x(n) is an element of A, with the Jacobian matrix (D-i(x(j))) invertible, [D-i, D-j] = 0, D-i(p) = cD(i), c is an element of {0,1}, 1 <= i, j <= n, we obtain elements y(1), ... , y(n) is an element of A, such that D-i(y(j)) = delta(ij) (1 + cy(i)), using properties of H-c-Galois extensions. A concrete structure theorem for a commutative k-algebra A, as a free module on the subring A(delta) of A consisting of the coinvariant elements with respect to 5, is proved in the additive case.