Definition:Ring of Formal Laurent Series

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 $\left({R((X)), \lambda \circ \kappa, \lambda(X)}\right)$.

Source of Name

This entry was named for Pierre Alphonse Laurent.

Also see

• Ring of Formal Laurent Series over Field is Field
• Definition:Polynomial Ring
• Definition:Ring of Laurent Polynomials
• Definition:Ring of Formal Power Series