Formal power series ring

Let $R$ be a commutative ring with $1$ . The formal power series ring $R[[x]]$ consists of all coefficient sequences $(a_0,a_1,\ldots)$ , written

\sum_{i=0}^{\infty}a_i x^i,\qquad a_i\in R.

Two series are equal exactly when every coefficient agrees. Addition is coefficientwise, and multiplication is the Cauchy product

\left(\sum_{i\geq 0}a_i x^i\right) \left(\sum_{j\geq 0}b_j x^j\right) = \sum_{n\geq 0}\left(\sum_{i+j=n}a_i b_j\right)x^n.

Each coefficient on the right is a finite sum, so multiplication is algebraically defined without asking the series to converge numerically.

Augmentation and topology

Evaluation at $x=0$ is the augmentation

\epsilon:R[[x]]\longrightarrow R,\qquad \sum a_i x^i\longmapsto a_0,

whose kernel is the principal ideal $(x)$ . The powers $(x^N)$ define the $x$ -adic topology, and truncation gives

R[[x]]\cong\varprojlim_N R[x]/(x^N).

Thus $R[[x]]$ is complete and separated for its $x$ -adic topology. This topology, rather than analytic convergence, is what makes infinite algebraic manipulations meaningful. Consequently, formal power series rings are fundamental examples in completion and deformation arguments.

Units and local structure

A series $f=\sum a_i x^i$ is a unit precisely when $a_0$ is a unit of $R$ . The coefficients of $f^{-1}$ are then determined recursively from $ff^{-1}=1$ .

There is a natural inclusion of polynomials into $R[[x]]$ as the finite series. If $R$ is a field, $R[[x]]$ is a local ring with unique maximal ideal $(x)$ . For general $R$ , maximal ideals are obtained from maximal ideals $\mathfrak m\subset R$ as $\mathfrak mR[[x]]+(x)$ .

Examples

For a field $k$ , $k[[x]]$ is a domain in which $x$ is nonzero but topologically “small”.
In $\mathbb{Z}[[x]]$ , the element $1+x$ is a unit with inverse $1-x+x^2-x^3+\cdots$ .
An expression with infinitely many negative powers of $x$ is not in $R[[x]]$ .

Several variables and substitution

The ring $R[[x_1,\ldots,x_n]]$ is the several-variable analogue. Infinite substitution is well-defined when the substituted series have zero constant term; this is a topological condition, not an analytic one.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Appendix A, power-series rings.
Nicolas Bourbaki, Algebra II: Chapters 4–7, Springer, 1990. Relevant: Chapter 4, formal series and restricted substitutions.