A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if f and k satisfy the conditions above and g is a smooth function near the origin, then we can write where q and r are smooth, and as a function of t, r is a polynomial of degree less than k. This means that for some smooth functions rj(x).
The obvious way of doing this does not work: although smooth functions have a formal power series expansion at the origin, and the Weierstrass preparation theorem applies to formal power series, the formal power series will not usually converge to smooth functions near the origin.
For a proof along these lines see (Mather 1968) or (Hörmander 1983a, section 7.5) The Malgrange preparation theorem can be restated as a theorem about modules over rings of smooth, real-valued germs.
If X is a manifold, with p∈X, let C∞p(X) denote the ring of real-valued germs of smooth functions at p on X.
Let Mp(X) denote the unique maximal ideal of C∞p(X), consisting of germs which vanish at p. Let A be a C∞p(X)-module, and let f:X → Y be a smooth function between manifolds.