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 $\tuple {\map R {\paren X}, \iota, X}$ 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:

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 $\tuple {\map R {\paren X}, \lambda \circ \kappa, \map \lambda X}$.

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