FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 3, PAGES 151-224

**Laurent rings**

D. A. Tuganbaev

Abstract

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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 $A((x;$f))
and formal pseudo-differential operator rings
$A((t$-1; d)),
with multiplication twisted by either an
automorphism $$f or
a derivation $$d 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.

Location: http://mech.math.msu.su/~fpm/eng/k06/k063/k06309h.htm

Last modified: July 22, 2006