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

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

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

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