Definition:Ring of Formal Power Series

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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 $\struct {R \sqbrk X, \iota, X}$ where:

$R \sqbrk X$ is a commutative ring with unity
$\iota : R \to R \sqbrk X$ is a unital ring homomorphism, called canonical embedding
$X$ is an element of $RR \sqbrk X$, called indeterminate

that may be defined as follows:

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

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

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

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

The ring of formal power series over $R$ is the ordered triple $\struct {R \sqbrk \N, \iota, X}$

Multiple variables

Also see