On the weierstrass preparation theorem
Matthew J. O’Malley · Rocky Mountain Journal of Mathematics · 1972
Introduction.Suppose that R is a commutative ring with identity, X is an indeterminate over ft, and S = ft [ [X]] is the formal power series ring.In [1, §3, Proposition 6], the following result (Weierstrass Preparation Theorem) is proved when ft is a local ring, complete in its maximal ideal adic topology: Suppose that /= ^i^aiX 1 G S, where, for some n § 1, a" is a unit of ft and (a 0 , fli, * ' ", a n -i) Q M, the maximal ideal of ft.Then there exists a unique pair u, F G S such that u is a unit of S and F is a monic polynomial of degree n with the property that the coefficients of X* in F, for i there exists a u G o>, depending on n and m, such that A^ Ç A^ fi A™.