Formal power series ring

The ring R[[x]] of infinite power series with coefficients in R and Cauchy product.

Formal power series ring

Let $R$ be a commutative ring with $1$ . The formal power series ring $R[[x]]$ consists of all infinite sums

\sum_{i=0}^{\infty} a_i x^i \quad (a_i\in R),

with addition defined coefficientwise and multiplication defined by the Cauchy product (so the coefficient of $x^n$ is $\sum_{i+j=n} a_i b_j$ ).

There is a natural inclusion of polynomials into $R[[x]]$ (finite series). If $R$ is a field, then $R[[x]]$ is a local ring with unique maximal ideal generated by $x$ , and it is a fundamental example for completions and deformation arguments.

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 $R[[x]]$ .