Fatou extension

of a commutative ring

A commutative ring containing such that each formal power series which is -rational is in fact -rational. Recall that a formal power series is -rational, a commutative ring, if there exist two polynomials such that and , that is, is equal to the formal expansion of . For instance, if is a field extension (cf. also Extension of a field), then is a Fatou extension of .

Fatou extensions are well characterized in the integral case. Thus, from now on, is supposed to be an integral domain with quotient field . The example above shows that an integral domain containing is a Fatou extension of if and only if the ring is a Fatou extension of . If the integral domain is Noetherian (cf. Noetherian ring), then its quotient field is a Fatou extension of , and, hence, every integral domain containing is a Fatou extension of . Many rings are Noetherian: for instance, every finitely generated -algebra is Noetherian.

For a rational function , there are several representations of the form with . Such a representation is said to be:

a) unitary if the non-zero coefficient of corresponding to the lowest degree is ;

b) irreducible if and are relatively prime in (cf. also Mutually-prime numbers);

c) with coefficients in if . Let denote the set of rational functions with a unitary representation with coefficients in , and let denote the set of Laurent power series, that is,

(these notations extend the classical notations and ).

To say that the integral domain is a Fatou extension of is nothing else than to write:

in other words, each rational function , where denotes the quotient field of , which has a unitary representation with coefficients in and a Laurent expansion at with coefficients in , has a unitary representation with coefficients in .

A rational function has a unique unitary and irreducible representation. With respect to this representation, there are two main results:

1) The ring is the set of elements of which admit a unitary and irreducible representation whose coefficients are integral over .

2) For every element of , the coefficients of the unitary and irreducible representation are almost integral over . Recall that an element of is almost integral over if there exists a non-zero element of such that belongs to for each positive integer . Each element of which is integral over is almost integral over .

An integral domain containing is a Fatou extension of if and only if each element of which is both integral over and almost integral over is integral over [a1]. The Noetherian case considered above follows from the fact that if is Noetherian, then each element of which is almost integral over is integral over .

The definition of Fatou extension may be easily extended to semi-ring extensions. Then, is a Fatou extension of , while is not a Fatou extension of , nor of [a2].

Moreover, the notion may be considered for formal power series in non-commuting variables, which have applications in system and control theory [a3]. It turns out that the previous characterization in the integral case still holds.

References

[a1]	P.-J. Cahen, J.-L. Chabert, "Eléments quasi-entiers et extensions de Fatou" J. Algebra , 36 (1975) pp. 185–192
[a2]	J. Berstel, C. Reutenauer, "Rational series and their languages" , Springer (1988)
[a3]	A. Salomaa, M. Soittola, "Automata-theoretic aspects of formal power series" , Springer (1978)