# Definition:Ring of Formal Laurent Series

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## Definition

Let $R$ be a commutative ring with unity.

### One variable

A **ring of formal Laurent series in one variable** over $R$ is a pointed algebra over $R$, that is, an ordered triple $\tuple {\map R {\paren X}, \iota, X}$ where:

- $\map R {\paren X}$ is a commutative ring with unity
- $\iota: R \to \map R {\paren X}$ is a unital ring homomorphism, called canonical embedding
- $X$ is an element of $\map R {\paren X}$, called variable

that may be defined as follows:

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Let $\tuple {R \sqbrk {\sqbrk X}, \kappa, X}$ be a ring of formal power series in one variable over $R$.

Let $\tuple {\map R {\paren X}, \lambda}$ be the localization of $R \sqbrk {\sqbrk X}$ at $X$.

The **ring of formal Laurent series** over $R$ is the ordered triple $\tuple {\map R {\paren X}, \lambda \circ \kappa, \map \lambda X}$.

### Multiple variables

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## Source of Name

This entry was named for Pierre Alphonse Laurent.