# Definition:Ring of Formal Power Series

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## 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 $\left({R[[X]], \iota, X}\right)$ where:

- $R[[X]]$ is a commutative ring with unity
- $\iota : R \to R[[X]]$ is a unital ring homomorphism, called canonical embedding
- $X$ is an element of $R[[X]]$, 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 $\left({R[[\N]], \iota, X}\right)$