Laurent rings

D. A. Tuganbaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 3, pp. 151-224

Résumé

This is a study of ring-theoretic properties of a Laurent ring over a ring $A$, which is defined to be any ring formed from the additive group of Laurent series in a variable $x$ over $A$, such that left multiplication by elements of $A$ and right multiplication by powers of $x$ obey the usual rules, and such that the lowest degree of the product of two nonzero series is not less than the sum of the lowest degrees of the factors. The main examples are skew-Laurent series rings $A((x;\varphi))$ and formal pseudo-differential operator rings $A((t^{-1};\delta))$, with multiplication twisted by either an automorphism $\varphi$ or a derivation $\delta$ of the coefficient ring $A$ (in the latter case, take $x=t^{-1}$). Generalized Laurent rings are also studied. The ring of fractional $n$-adic numbers (the localization of the ring of $n$-adic integers with respect to the multiplicative set generated by $n$) is an example of a generalized Laurent ring. Necessary and/or sufficient conditions are derived for Laurent rings to be rings of various standard types. The paper also includes some results on Laurent series rings in several variables.