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,
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(these notations extend the classical notations and
).
To say that the integral domain is a Fatou extension of
is nothing else than to write:
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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) |
Fatou extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fatou_extension&oldid=18283