Category:Definitions/Formal Laurent Series

This category contains definitions related to Formal Laurent Series.
Related results can be found in Category:Formal Laurent Series.

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