Power Series Extension Research Articles

On annihilator ideals in skew power series rings

The purpose of this paper is to study the behavior of the IN and SA property with respect to the skew power series and Laurent series extensions. For an α-compatible ring R it is shown that the skew power series ring is right SA if and only if R is right SA and satisfies the (SQA2) condition. Also it is shown that a semiprime ring R is right SA if and only if is right SA. Moreover, if α is an automorphism, then we have similar results for skew Laurent series ring Furthermore, if R is a reduced left Noetherian ring, then R is left IN if and only if is left IN if and only if is left IN. Also we present some examples to show that if we relax each assumption on R, our results can fail.

Power Series Extensions of Certain Functions of a Real Variable

The domain of the function $ f(x)=\cos \sqrt{x} $ is the set of all nonnegative real numbers. In this article, we will show how to use power series to extend this function to an analytic function defined on the entire real line. We will then show how this analytic extension of $ f(x) $ makes it easier and quicker for calculus students to compute derivatives of $ f(x) $ at the origin. We will moreover describe the process of extending the domain of any function of the form $ g(\sqrt{x}) $ for a given even analytic function $ g(x) $.

On S-GCD domains

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).

Noetherian-like properties in polynomial and power series rings

There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions R[X1〛⋯[Xn〛 over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that R/P is integrally closed for each prime ideal P of R, then R[X] is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then R[X1,⋯,Xn] is an SFT-ring. Finally, we show that the power series extension R〚X〛 over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian.

Differential Extensions of Weakly Principally Quasi-Baer Rings

A ring R is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo, the right annihilator of any principal right ideal, is flat. We study the relationship between the weakly p.q.-Baer property of a ring R and those of the differential polynomial extension R[x;δ], the pseudo-differential operator ring R((x− 1;δ)), and also the differential inverse power series extension R[[x− 1;δ]] for any derivation δ of R. Examples to illustrate and delimit the theory are provided.

LCM-stability and formal power series

In this paper we study the LCM-stability property and other related concepts, and their universality in the case of polynomial and formal power series extensions.

EXTENSIONS OF Σ-ZIP RINGS

t. In this note we consider a new concept, so called Σ-zip ring, which unifies zip rings and weak zip rings. We observe the basic properties of Σ-zip rings, constructing typical examples. We study the relationship between the Σ-zip property of a ring R and that of its Ore extensions and skew generalized power series extensions. As a consequence, we obtain a generalization of several known results relating to zip rings and weak zip rings.

A power series revisit of the PBE exchange density-functional approximation: The PBEpow model.

PBEpow is a power series extension of the Perdew-Burke-Ernzerhof (PBE) exchange generalized gradient approximation. This novel variant is constructed by an inductive determination of the power series coefficients to fulfill physical fundamental constants in the slowly and rapidly varying density limits. When combined with the PBE correlation, PBEpow significantly improves the estimation of molecular covalent interactions such as energies of atomization reactions and keeps the line of performance of PBE for noncovalent interactions. More generally this work provides new insights about how to improve the modeling of short- and long-range interactions at the generalized gradient approximation level without the help of any kind of empirical parameterization.

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.

The piecewise Noetherian property in power series rings over a valuation domain

Let V be a valuation domain and let P be a nonzero prime ideal of V. We characterize when V〚X〛P〚X〛 is a valuation domain and when it is a Noetherian ring. We then show that V〚X〛 is piecewise Noetherian if and only if V is Noetherian. As a corollary, we obtain that the piecewise Noetherian property is not preserved under the power series extension.

The Ohm–Rush content function

The content of a polynomial over a ring R is a well-understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of R, although it can behave quite badly. We examine five properties an algebra may have with respect to this function — content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm–Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm–Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Prüfer domains.

Semi-Baer modules

In this paper, we introduce the concepts of semi-Baer, semi-quasi Baer, semi-p.q. Baer and semi-p.p. modules as a generalization of Baer, quasi Baer, p.q. Baer and p.p. modules respectively. To define these concepts, we introduce concepts of multiplicative order of an element and a multiplicatively finite element in rings. Further, we characterize these concepts in modules over reduced rings. Also, it is proved that semi-Baer and semi-quasi Baer properties are preserved by polynomial extensions and power series extensions of modules. It is proved that for a ring R and a monoid G, if the semi group ring RG is semi-Baer (semi-quasi Baer) then so is R.

PS-Modules over Ore Extensions and Skew Generalized Power Series Rings

A rightR-moduleMRis called a PS-module if its socle,SocMR, is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. LetRbe an associative ring with identity,MRa unitary rightR-module,O=Rx;α,δOre extension,MxOa rightO-module,S,≤a strictly ordered additive monoid,ω:S→EndRa monoid homomorphism,A=RS,≤,ωthe skew generalized power series ring, andBA=MS,≤RS,≤, ωthe skew generalized power series module. Then, under some certain conditions, we prove the following: (1) IfMRis a right PS-module, thenMxOis a right PS-module. (2) IfMRis a right PS-module, thenBAis a right PS-module.

Modules with the Beachy–Blair Condition

A right R-module satisfies the right Beachy–Blair condition if each of its faithful submodules is cofaithful. In this article we study the relationship between the right Beachy–Blair condition of a right R-module and its skew polynomial, skew monoid and skew generalized power series extensions. Consequently, several known results regarding rings satisfying the right Beachy–Blair condition are extended to a more general setting.

RINGS WITH THE BEACHY–BLAIR CONDITION

A ring satisfies the left Beachy–Blair condition if each of its faithful left ideal is cofaithful. Every left zip ring satisfies the left Beachy–Blair condition, but both properties are not equivalent. In this paper we will study the similarities and the differences between zip rings and rings with the Beachy–Blair condition. We will also study the relationship between the Beachy–Blair condition of a ring and its skew polynomial and skew power series extensions. We give an example of a right zip ring that is not left zip, proving that the zip property is not symmetric.

Integral expressions for Hilbert-type infinite multilinear form and related multiple Hurwitz-Lerch Zeta functions

The article deals with different kinds integral expressions concerning multiple Hurwitz-Lerch Zeta function (introduced originally by Barnes ), Hilbert-type infinite multilinear form and its power series extension. Here Laplace integral forms and multiple Mellin-Barnes type integral representation are derived for these special functions. As a special cases of our investigations we deduce the integral expressions for the Matsumoto's multiple Mordell-Tornheim Zeta function, that is, for Tornheim's double sum i.e. Mordell-Witten Zeta, for the multiple Hurwitz Zeta and for the multiple Hurwitz-Euler Eta function, recently studied by Choi and Srivastava .

QFT on homothetic Killing twist deformed curved spacetimes

We study the quantum field theory (QFT) of a free, real, massless and curvature coupled scalar field on self-similar symmetric spacetimes, which are deformed by an abelian Drinfel'd twist constructed from a Killing and a homothetic Killing vector field. In contrast to deformations solely by Killing vector fields, such as the Moyal-Weyl Minkowski spacetime, the equation of motion and Green's operators are deformed. We show that there is a *-algebra isomorphism between the QFT on the deformed and the formal power series extension of the QFT on the undeformed spacetime. We study the convergent implementation of our deformations for toy-models. For these models it is found that there is a *-isomorphism between the deformed Weyl algebra and a reduced undeformed Weyl algebra, where certain strongly localized observables are excluded. Thus, our models realize the intuitive physical picture that noncommutative geometry prevents arbitrary localization in spacetime.

Krull dimension of mixed extensions

For a commutative ring R with identity, dim R shall stand for the Krull dimension of R . It is known that dim R [ x ] ≤ 2 dim R + 1 . We show that this does not hold for the power series extensions. Using mixed extensions, we construct an example of a finite-dimensional integral domain R such that 2 dim R + 1 < dim R 〚 x 〛 < ∞ . Let D be a finite-dimensional SFT Prüfer domain and D [ x 1 〛 ⋯ [ x n 〛 be a mixed extension. According to Arnold, dim D [ x 1 , … , x n ] = dim D + n and dim D 〚 x 1 , … , x n 〛 = n dim D + 1 . We generalize Arnold’s result by showing that dim D [ x 1 〛 ⋯ [ x n 〛 = n dim D + 1 provided that there is at least one power series extension. In particular, if R = D [ x 1 , ⋯ , x n − 1 ] and dim D = d > 2 ( n − 1 ) / ( n − 2 ) , then dim R 〚 x 〛 = d n + 1 > 2 dim R + 1 . This is an answer to the question of Coykendall and Gilmer.

Noetherian Skew Inverse Power Series Rings

We study skew inverse power series extensions R[[y− 1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ-derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension are all equal to 2n.

ON REDUCED MODULES

Let α be an endomorphism of an arbitrary ring R with identity. In this note, we concern the relations between polynomial and power series extensions of a reduced module. Among others we prove that a ring R is α- reduced if and only if every áat right R-module is -reduced, and for a module M, M[x] is α-reduced if and only if M[x; x1 ] is α-reduced.