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A formal power series with coefficients in a ring is called a formal power series over . The formal power series over a ring R {\displaystyle R} form a ring, commonly denoted by R [ [ x ] ] . {\displaystyle R[[x]].} (It can be seen as the ( x ) -adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p ...
The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be ...
A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).
pronounced "R I hat". The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I. Thus π is injective if and only if this intersection reduces to the zero element of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a ...
Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows. As an R-module, H is free with a basis 1 = D (0), D (1), D (2), ... The coproduct Δ is given by ΔD (n) = ΣD (i) ⊗ D (n−i) (so the dual of this coalgebra is just the ring of formal power series).
Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product.
Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form (), []: the ring of Gaussian integers, [2] [] (where is a primitive cube root of 1): the Eisenstein integers,
The ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0.