Taylor expansion of noncommutative polynomials

The algebra A = K[x(1),...,x(n)] freely generated by a system x(1),..., x(n) of variables over a commutative field K is canonically isomorphic with the algebra of skew polynomials over a free algebra A(0) over K with respect to a system of derivations on A(0) and a skew n x n-matrix over A(0). If char K = 0, then A(0) is the algebra of polynomials f is an element of A where partial derivatives partial derivative f/partial derivative x(i) vanish far all i. One obtains for any f is an element of A an expansion f = Sigma a(nu)x(nu) with a(nu) is an element of A(0). It is suggested to call it the right Taylor expansion of f. This result is applied to extend a theorem of Bokhut and Makar-Limanov about free metabelian algebras to the case of finite characteristics.