Editing Weierstrass preparation theorem (section)

==Formal power series in complete local rings==

There is an analogous result, also referred to as the Weierstrass preparation theorem, for the ring of [[formal power series]] over [[complete local ring]]s ''A'':<ref>{{citation|author=Nicolas Bourbaki|author-link=Nicolas Bourbaki|title=Commutative algebra|location=chapter VII, §3, no. 9, Proposition 6|year=1972|publisher=Hermann}}</ref> for any power series <math>f = \sum_{n=0}^\infty a_n t^n \in A[[t]]</math> such that not all <math>a_n</math> are in the [[maximal ideal]] <math>\mathfrak m</math> of ''A'', there is a unique [[unit (ring theory)|unit]] ''u'' in <math>A[[t]]</math> and a polynomial ''F'' of the form <math>F=t^s + b_{s-1} t^{s-1} + \dots + b_0</math> with <math>b_i \in \mathfrak m</math> (a so-called distinguished polynomial) such that
:<math>f = uF.</math>
Since <math>A[[t]]</math> is again a complete local ring, the result can be iterated and therefore gives similar factorization results for formal power series in several variables.

For example, this applies to the ring of integers in a [[p-adic]] field. In this case the theorem says that a power series ''f''(''z'') can always be uniquely factored as π<sup>''n''</sup>·''u''(''z'')·''p''(''z''), where ''u''(''z'') is a unit in the ring of power series, ''p''(''z'') is a [[distinguished polynomial]] (monic, with the coefficients of the non-leading terms each in the maximal ideal), and π is a fixed [[uniformizer]].

An application of the Weierstrass preparation and division theorem for the ring <math>\mathbf Z_p[[t]]</math> (also called [[Iwasawa algebra]]) occurs in [[Iwasawa theory]] in the description of finitely generated modules over this ring.<ref>{{citation|author=Lawrence Washington|author-link=Lawrence C. Washington|title=Introduction to cyclotomic fields|publisher=Springer|year=1982|location=Theorem 13.12}}</ref>

There exists a non-commutative version of Weierstrass division and preparation, with ''A'' being a not necessarily commutative ring, and with formal skew power series in place of formal power series.<ref>{{cite journal |author=Otmar Venjakob |title=A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory |journal=J. Reine Angew. Math. |year=2003 |volume=2003 |issue=559 |pages=153–191 |doi=10.1515/crll.2003.047 |s2cid=14265629 |url=https://www.degruyter.com/document/doi/10.1515/crll.2003.047/pdf |accessdate=2022-01-27|arxiv=math/0204358 }} Theorem 3.1, Corollary 3.2</ref>