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 in {0,1}, the Lie Hopf algebra of the additive group and the multiplicative group on A, respectively. If there exist x_1, dots, x_n in A, with the Jacobian matrix (D_i(x_j)) invertible, [D_i,D_j] = 0, D_i^p = cD_i, c in {0, 1}, 1 leq i, j leq n, we obtain elements y_1,..., y_n in 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 delta, is proved in the additive case.