  We suggest the necessary/sufficient criteria for the existence of a
(order-by-order) solution y(x) of a functional equation F(x,y)=0 over a ring.
In full generality, the criteria hold in the category of filtered groups, this
includes the wide class of modules over (commutative, associative) rings. The
classical implicit function theorem and its strengthening obtained by Tougeron
and Fisher appear to be (weaker) particular forms of the general criterion.
  We obtain a special criterion for solvability of the equations arising from
group actions, g(w)=w+u, here u is "small". As an immediate application we
re-derive the classical criteria of determinacy, in terms of the tangent space
to the orbit.
  Finally, we prove the Artin-Tougeron-type approximation theorem: if a system
of C^\infty-equations has a formal solution and the derivative satisfies a
Lojasiewicz-type condition then the system has a C^\infty-solution.
