Left Skew Laurent Series

• Left Skew Laurent Series is a formal Laurent expansion with twisted multiplication (x·a = σ(a)x) that presents elements in a left-normal form.
• It embeds within generalized power series frameworks by enforcing artinian support conditions, thereby unifying left ideal theory with skew derivation methods.
• Applications include noncommutative Dedekind domain analysis, coding theory, and division ring constructions, demonstrating strong structural invariants and one-sided module properties.

Searching arXiv for recent and relevant papers on left/skew Laurent series and adjacent generalized series frameworks. Left skew Laurent series are formal Laurent expansions equipped with a twisted, generally noncommutative multiplication. In the automorphism-based setting, an element is written as $\sum_{i=k}^{\infty} a_i x^i$, with finitely many negative powers and an arbitrary infinite nonnegative tail, and multiplication is governed by a rule such as $xa=\sigma(a)x$. In the recent literature, the phrase has two closely related uses: it can mean the left-normal or left-ideal viewpoint on an ordinary skew Laurent series ring, and it can also denote a genuinely left skew Laurent construction built from a left skew derivation $(\sigma,\delta)$ (Vitas, 2024, Danchev et al., 27 Apr 2025, Gómez-Torrecillas et al., 7 Jul 2025).

1. Basic algebraic construction

In the formal skew Laurent series ring over a ring $D$ with automorphism $\sigma$, the underlying set is

$$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$

The defining features are that negative powers occur only finitely often, while nonnegative powers may continue indefinitely. Addition is termwise. Multiplication is twisted by

$xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$

so for monomials

$(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$

and for two series

$$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$

their product is

$$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$

This is the standard skew analogue of the ordinary Laurent series ring $xa=\sigma(a)x$0 (Vitas, 2024).

A broader formulation places skew Laurent series inside a skew generalized power series ring $xa=\sigma(a)x$1, where $xa=\sigma(a)x$2 is a strictly totally ordered monoid and $xa=\sigma(a)x$3 is a monoid homomorphism. Elements are functions $xa=\sigma(a)x$4 with support that is artinian and narrow, and multiplication is given by the twisted convolution

$xa=\sigma(a)x$5

When $xa=\sigma(a)x$6 with its usual order, the support condition becomes “bounded below,” so the admissible elements are exactly series of the form $xa=\sigma(a)x$7. In this framework the basic skew relation is

$xa=\sigma(a)x$8

which specializes to $xa=\sigma(a)x$9 in the Laurent case (Danchev et al., 27 Apr 2025).

2. What “left” means

One important usage of “left skew Laurent series” does not name a different ring. In the formal skew Laurent series ring $(\sigma,\delta)$0, the usual presentation is

$(\sigma,\delta)$1

with coefficients on the left. The same element can also be written in left normal form as

$(\sigma,\delta)$2

For left-sided arguments, the relevant coefficient datum is the left constant term

$(\sigma,\delta)$3

and the paper on Dedekind-domain coefficients states explicitly that the left-sided theory is symmetric to the right-sided one, except that one replaces $(\sigma,\delta)$4 by $(\sigma,\delta)$5. In that sense, “left skew Laurent series” can mean the left-normal-form and left-ideal bookkeeping for the same underlying ring (Vitas, 2024).

A second usage is more structural. In the generalized-power-series literature, the multiplication law

$(\sigma,\delta)$6

places the twist on the coefficient coming from the right factor, while coefficients are still written on the left of monomials. The monomial relation

$(\sigma,\delta)$7

therefore already encodes a left-oriented skew convention (Danchev et al., 27 Apr 2025).

The phrase becomes fully explicit in the left-skew-derivation setting. For a left skew derivation $(\sigma,\delta)$8, the Ore extension $(\sigma,\delta)$9 is defined by

$D$0

The later Laurent theory built from this rule is described in that paper as a theory of left skew formal power series and left skew Laurent formal power series (Gómez-Torrecillas et al., 7 Jul 2025).

3. Generalized, partial, and derivation-based forms

The generalized power series construction encompasses skew Laurent polynomial rings, skew Laurent series rings, skew monoid rings, skew group rings, and skew Malcev–Neumann series rings as special cases. In the left APP-ring framework, the ring

$D$1

consists of functions $D$2 whose support is artinian and narrow, with multiplication

$D$3

The embedded coefficient and monomial elements satisfy

$D$4

For $D$5 and $D$6, the resulting elements are precisely

$D$7

with relation $D$8. In that setting the construction is exactly a left skew Laurent series ring in the sense of left coefficient presentation and left twisting (Zhao, 2010).

A more general non-global version is the twisted partial skew Laurent series ring $D$9, attached to a unital twisted partial action of $\sigma$0 on $\sigma$1. Its elements are

$\sigma$2

and homogeneous multiplication is

$\sigma$3

This extends ordinary Laurent series, classical skew Laurent series from a global automorphism, and twisted global skew Laurent series (Cortes et al., 2017).

The left-skew-derivation theory goes further. If $\sigma$4 is locally nilpotent, then there exists a unique reasonable ring structure on $\sigma$5 containing $\sigma$6 as a subring, denoted $\sigma$7. If moreover $\sigma$8 is an automorphism and

$\sigma$9

is nilpotent, then one can localize at $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$0 and obtain the ring of left skew Laurent formal power series

$$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$1

In that ring,

$$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$2

where $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$3 is such that $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$4 (Gómez-Torrecillas et al., 7 Jul 2025).

4. Ideal theory and structural invariants

Over a commutative Dedekind domain $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$5 with automorphism $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$6, the formal skew Laurent series ring

$$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$7

is shown to be a noncommutative Dedekind domain. In the framework of that paper, this means that $$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$8 is a domain, is left and right noetherian, is left and right hereditary, and is an Asano order. The same paper proves the simplicity criterion

$$D((x;\sigma))=\left\{\sum_{i=k}^{\infty} a_i x^i \mid a_i\in D,\ k\in\mathbb Z\right\}.$$9

It also introduces the constant ideal $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$0 of a right ideal $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$1 and proves the extension proposition

$xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$2

for $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$3. The paper explicitly notes that not every right ideal is literally of the form $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$4; over $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$5, $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$6 is an example. Under the hypothesis that $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$7 acts trivially on the ideal class group $xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$8, one has

$xa=\sigma(a)x,\qquad x^i a=\sigma^i(a)x^i,$9

and any two stably isomorphic finitely generated projective $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$0-modules are isomorphic. For $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$1 not a field, it also computes

$(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$2

and, when $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$3 acts trivially on $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$4,

$(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$5

(Vitas, 2024).

In the twisted partial setting, primeness and radical theory are controlled by $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$6-invariant ideals of the coefficient ring. The paper proves

$(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$7

and identifies the prime radical by

$(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$8

It also shows that if $(ax^m)(bx^n)=a\,\sigma^m(b)\,x^{m+n},$9 is semiprime, then $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$0 is semiprime, and for semiprime $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$1,

$$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$2

(Cortes et al., 2017).

5. One-sided finiteness, cancellation, and module-theoretic properties

Several papers study genuinely one-sided properties of skew Laurent-type rings. In the skew generalized power series setting, if $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$3 is $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$4-compatible, $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$5 is a strictly totally ordered monoid, and $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$6 is a monoid homomorphism, then Theorem 2.9 states that if $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$7 is abelian and semi-regular with $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$8 nilpotent, $$f=\sum_{i=k}^{\infty} a_i x^i,\qquad g=\sum_{j=\ell}^{\infty} b_j x^j,$$9 is $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$0-McCoy. Its Laurent-series specialization is Corollary 2.11: if $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$1 is a compatible automorphism of $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$2 and $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$3 is abelian and semi-regular with $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$4 nilpotent, then

$$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$5

is skew McCoy, hence both left and right McCoy (Danchev et al., 27 Apr 2025).

The left APP-ring paper gives a criterion tailored to left annihilator behavior. If $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$6 is a strictly totally ordered monoid, $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$7 is a monoid homomorphism, and $$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$8 satisfies descending chain condition on right annihilators, then

$$fg=\sum_{n=k+\ell}^{\infty}\left(\sum_{i+j=n} a_i\,\sigma^i(b_j)\right)x^n.$$9

is left APP if and only if for any $xa=\sigma(a)x$00-indexed subset $xa=\sigma(a)x$01 of $xa=\sigma(a)x$02,

$xa=\sigma(a)x$03

is right $xa=\sigma(a)x$04-unital. For $xa=\sigma(a)x$05, this is the natural left APP criterion for left skew Laurent series in the ordered generalized-power-series sense (Zhao, 2010).

On the Euclidean side, the skew Laurent formal series ring

$xa=\sigma(a)x$06

is shown to inherit right $xa=\sigma(a)x$07-Euclideanity from $xa=\sigma(a)x$08; with a multiplicative norm it becomes a right principal ideal domain, and under stronger hypotheses it has elementary reduction of matrices. The paper explicitly remarks that for left $xa=\sigma(a)x$09-Euclidean domains, the left-side version of its basic division proposition is also valid, but it does not develop a full left-sided theory of all the later theorems (Romaniv et al., 2017).

The generalized-power-series literature also records left/right chain conditions and Noetherianity criteria. One paper states as a consequence that power series rings, Laurent series rings, skew power series rings, skew Laurent series rings, and generalized power series rings are reduced and satisfy the ascending chain condition on principal left or right ideals under its hypotheses (Padashnik et al., 2016). Another proves that $xa=\sigma(a)x$10 is left Noetherian if and only if $xa=\sigma(a)x$11 is left Noetherian and $xa=\sigma(a)x$12 is finitely generated, but it also notes that this framework is closer to ordered generalized or Mal'cev–Neumann Laurent series than to arbitrary bilateral skew Laurent series (Padashnik et al., 2016).

A further right-sided structural result concerns semidistributivity. If $xa=\sigma(a)x$13 is a right semidistributive semilocal ring, then $xa=\sigma(a)x$14 is a right semidistributive right Artinian ring, and $xa=\sigma(a)x$15 is right Artinian. The same paper identifies

$xa=\sigma(a)x$16

so a plausible implication is that left-sided analogues should be read through opposite rings rather than by changing the underlying Laurent construction (Tuganbaev, 2020).

6. Division rings, applications, and terminological boundaries

Over a field $xa=\sigma(a)x$17 with automorphism $xa=\sigma(a)x$18, the skew Laurent series division ring

$xa=\sigma(a)x$19

consists of series $xa=\sigma(a)x$20 with finitely many negative terms and multiplication $xa=\sigma(a)x$21. In that setting, every element is shown to be a product of two additive commutators. This result depends heavily on the skew Laurent series expansion and on explicit identities such as

$xa=\sigma(a)x$22

when $xa=\sigma(a)x$23 (Jang et al., 15 Jan 2025).

At the rational rather than formal-series level, skew Laurent polynomial rings over firs,

$xa=\sigma(a)x$24

are shown to admit universal division rings of fractions of the form

$xa=\sigma(a)x$25

with matrix invertibility governed by full or stably full matrices according to whether the ring is Sylvester or pseudo-Sylvester. This provides a fraction-theoretic backdrop for Laurent-type skew extensions, although it is not a theory of left skew Laurent series as formal series (Henneke et al., 2020).

A modern application is coding theory. The paper on cyclic convolutional codes emphasizes that when a skew derivation is introduced, serious difficulties arise in defining a skewed module structure on Laurent series, and it therefore develops a purely algebraic treatment of the left skew Laurent series built from a left skew derivation when possible (Gómez-Torrecillas et al., 7 Jul 2025).

The term should also be separated from several nearby but different usages of “Laurent series.” The field $xa=\sigma(a)x$26 studied in Diophantine approximation is an ordinary commutative Laurent-series field with no twisting automorphism and no left/right issue (Kalaydzhieva, 2020). “Laurent skew orthogonal polynomials” concern a skew-symmetric bilinear form on $xa=\sigma(a)x$27, not skew Laurent series rings (Miki, 2020). Multivariable algebraic Laurent series over $xa=\sigma(a)x$28 are studied through support cones and gap theorems in a commutative setting (Aroca et al., 2018). The noncommutative Laurent phenomenon for generalized Kontsevich automorphisms is about finite Laurent polynomials in a skew-field of rational expressions, not Laurent series in the Ore-extension sense (Rupel, 2017).

Taken together, these works show that left skew Laurent series are not a single rigid object but a family of closely related constructions. In the automorphism case, the decisive data are the left coefficient convention, the rule $xa=\sigma(a)x$29, and the passage between right-normal and left-normal forms. In the generalized and partial settings, ordered supports, partial actions, and cocycles control the Laurent expansion. In the skew-derivation setting, existence itself becomes conditional on local nilpotence and localization. Across these variants, the recurrent themes are one-sided ideal theory, localization at powers of the skew variable, and the persistence of strong structural invariants in a noncommutative Laurent environment.