Formal power series

over a ring in commuting variables

An algebraic expression of the form

where is a form of degree in with coefficients in . The minimal value of for which is called the order of the series , and the form is called the initial form of the series.

If

are two formal power series, then, by definition,

and

where

The set of all formal power series forms a ring under these operations.

A polynomial , where is a form of degree , is identified with the formal power series , where for and for . This defines an imbedding of the polynomial ring into . There is a topology defined on for which the ideals

form a fundamental system of neighbourhoods of zero. This topology is separable, the ring is complete relative to it, and the image of under the imbedding is everywhere dense in . Relative to this topology, a power series is the limit of its partial sums .

Suppose that is a commutative ring with an identity. Then so is . If is an integral domain, then so is . A formal power series is invertible in if and only if is invertible in . If is Noetherian, then so is . If is a local ring with maximal ideal , then is a local ring with maximal ideal .

If a local ring is separable and complete in the -adic topology, then the Weierstrass preparation theorem is true in . Let be a formal power series such that for some the form contains a term , where , and let be the minimal index with this property. Then , where is an invertible formal power series and is a polynomial of the form , where the coefficients belong to the maximal ideal of . The elements and are uniquely determined by .

The ring of formal power series over a field or a discretely-normed ring is factorial.

Rings of formal power series in non-commuting variables have also been studied.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , v. Nostrand (1960)

Comments

Power series in non-commuting variables are becoming rapidly more important and find applications in combinatorics (enumerative graph theory), computer science (automata) and system and control theory (representation of the input-output behaviour of non-linear systems, especially bilinear systems); cf. the collection [a1] for a first idea.

Let be a ring containing (or provided with a ring homomorphism ), let be an ideal in and suppose that is complete in the -adic topology on . Let be elements of . Then an expression

where the range over , , has a well-defined meaning in (as the unique limit of the finite sums

as ). Such an expression is also called a formal power series over . Mapping to , , defines a (continuous) homomorphism . If this homomorphism is injective, the are said to be analytically independent over .

Let now be a field with a multiplicative norm on it (i.e. ), e.g. with the usual norm or , the rational field, with the norm if , where is the -adic valuation on ( for is the exponent of the largest power of the prime number that divides ; ). Now consider all formal power series over such that there exists positive numbers and such that . These form a subring of , called the ring of convergent power series over and denoted by (or , but the latter notation also occurs for the ring of power series in non-commuting variables over ). The Weierstrass preparation theorem also holds in .

References

[a1]	J. Berstel (ed.) , Series formelles en variables noncommutatives et aplications , Lab. Inform. Théor. Programmation (1978)
[a2]	M. Nagata, "Local rings" , Interscience (1960)