Let A = k[x1,…,xn] be the algebra of polynomials in n variables over a field k of characteristic zero.The author obtained a characterization of k-endomorphisms of A satisfying the Jacobian condition as mapping irreducible polynomials to square-free ones.De Bondt and Yan proved that the condition of mapping square-free polynomials to square-free ones is also equivalent to the Jacobian condition.The above properties of an endomorphism can be expressed in terms of irreducible,resp.square-free,elements of its image.We will present a generalization of the mentioned properties for subalgebras generated by m algebraically independent polynomials,for an arbitrary m∈ {1,2,…,n}.We are going also to discuss a characteristic-free attempt and to present some motivations connected with rings of constants of derivations.