# 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 $\left({R((X)), \iota, X}\right)$ 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 variable

that may be defined as follows:

Let $(R[[X]], \kappa, X)$ be a ring of formal power series in one variable over $R$.

Let $(R((X)), \lambda)$ be the localization of $R[[X]]$ at $X$.

The **ring of formal Laurent series** over $R$ is the ordered triple $\left({R((X)), \lambda \circ \kappa, \lambda(X)}\right)$.

### Multiple variables

## Source of Name

This entry was named for Pierre Alphonse Laurent.