Definition:Ring of Formal Power Series

Definition
Let $R$ be a commutative ring with unity.

One variable
A ring of formal power series over $R$ is a pointed algebra over $R$, that is, an ordered triple $\left({RX, \iota, X}\right)$ where:
 * $RX$ is a commutative ring with unity
 * $\iota : R \to RX$ is a unital ring homomorphism, called canonical embedding
 * $X$ is an element of $RX$, called indeterminate

that may be defined as follows:

Let $\N$ be the additive monoid of natural numbers.

Let $R\N$ be the big monoid ring of $R$ over $\N$.

Let $\iota : R \to R\N$ be the embedding.

Let $X \in R\N$ be the mapping $X : \N \to R$ defined by $X(n) = 1$ if $n=1$ and $X(n) = 0$ otherwise.

The ring of formal power series over $R$ is the ordered triple $\left({R\N, \iota, X}\right)$

Also see

 * Definition:Polynomial Ring
 * Definition:Ring of Formal Laurent Series