In the present work, we study ring-theoretical properties of (skew-formal) Laurent series rings and rings of (formal) pseudo-differential operators; we also introduce the notion of a Laurent ring that generalizes the notions of skew-Laurent series rings and pseudo-differential operator rings. The use of skew-Laurent series rings was begun in the works of Schur, Dickson, and Hilbert at the beginning of XX century. For example, Hilbert used the skew-Laurent series ring in the study of the independence of geometry axioms (Hilbert has constructed a division ring which is infinite-dimensional over its center). The study of Laurent series rings with arbitrary coefficient ring was initiated by Lorenz [19], Risman [36], and Smits [39]. Laurent series rings are useful in ring theory. For example, Makar-Limanov has used skew-Laurent series rings in two variables to show that the ring of fractions of the Weyl algebra contains a free noncommutative subalgebra [20]. In the work of Goodearl and Small [10], Laurent series rings are used for the study of the Krull dimension and the global dimension of Noetherian P.I. rings.
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