View source for Fatou ring

{{TEX|done}}

An [[Integral domain|integral domain]] $A$ with quotient field $K$ such that if each rational function $R\in K(X)$ that has a Taylor expansion at $0$ with coefficients in $A$, has a unitary and irreducible representation with coefficients in $A$; that is, for each $R\in K(X)\cap A[[X]]$ there are $P,Q\in A[X]$ such that $R=P/Q$, $Q(0)=1$ and $P$ and $Q$ are relatively prime in $K[X]$. Fatou's lemma {{Cite|Fa}} states that $\mathbb{Z}$ is a Fatou ring.

Equivalently, $A$ is a Fatou ring means that if a sequence $\{ a_n\}_{n\in\mathbb{N}}$ of elements of $A$ satisfies a linear recursion formula

$$a_{n+s}+q_1a_{n+1}+\cdots+q_sa_n=0\textrm{ for }n\geq 0,$$

where $q_1,\ldots,q_s\in K$ and $s$ is as small as possible, then $q_1,\ldots,q_s\in A$.

If $A$ is a Fatou ring, then its quotient field $K$ is a [[Fatou extension|Fatou extension]] of $A$, but the converse does not hold. This is the reason why Fatou rings are sometimes called strong Fatou rings, while the domains $A$ such that $K$ is a Fatou extension of $A$ are called weak Fatou rings (in this latter case, every $R\in K(X)\cap A[[X]]$ has a unitary (not necessarily irreducible) representation with coefficients in $A$).

The coefficients of the unitary and irreducible representation of every element of $K(X)\cap A((X))$ are almost integral over $A$ (see [[Fatou extension|Fatou extension]]). An integral domain $A$ is a Fatou ring if and only if every element of $K$ which is almost integral over $A$ belongs to $A$ {{Cite|Ei}}; such domains are said to be completely integrally closed. For instance, a Noetherian domain (cf. also [[Noetherian ring|Noetherian ring]]) is completely integrally closed if and only if it is integrally closed (cf. also [[Integral domain|Integral domain]]). The rings of integers of number fields are completely integrally closed, and hence, Fatou rings.

The notion may be extended by considering [[Formal power series|formal power series]] in non-commuting variables. The characterization of this generalized property is still (1998) an open question.


====References====

* {{Ref|Ei}} S. Eilenberg, "Automata, languages and machines", '''A''', Acad. Press  (1974)

* {{Ref|Fa}} P. Fatou, "Sur les séries entières à coefficients entiers"  ''C.R. Acad. Sci. Paris Ser. A'', '''138'''  (1904)  pp. 342–344 {{ZBL|35.0257.02}}