PF-rings of generalised power series

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

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On semilocal, Bézout and distributive generalized power series rings

International Journal of Algebra and Computation ◽

10.1142/s0218196715500174 ◽

2015 ◽

Vol 25 (05) ◽

pp. 725-744 ◽

Cited By ~ 1

Author(s):

Ryszard Mazurek ◽

Michał Ziembowski

Keyword(s):

Power Series ◽

Neumann Series ◽

Polynomial Rings ◽

Group Rings ◽

Power Series Ring ◽

Generalized Power Series ◽

Series Ring ◽

Ordered Monoid ◽

Power Series Rings ◽

Totally Ordered Monoid

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

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McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500837 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1350083 ◽

Cited By ~ 6

Author(s):

A. ALHEVAZ ◽

D. KIANI

Keyword(s):

Power Series ◽

Commutative Ring ◽

Commutative Rings ◽

Laurent Polynomials ◽

Property A ◽

Power Series Ring ◽

Series Ring ◽

Monoid Ring ◽

Power Series Rings ◽

Base Ring

One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

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RADICALS OF SKEW GENERALIZED POWER SERIES RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501290 ◽

2012 ◽

Vol 12 (01) ◽

pp. 1250129 ◽

Cited By ~ 2

Author(s):

A. R. NASR-ISFAHANI

Keyword(s):

Power Series ◽

Power Series Ring ◽

Generalized Power Series ◽

Series Ring ◽

Ordered Monoid ◽

Monoid Homomorphism ◽

Property Transfer ◽

Power Series Rings ◽

Armendariz Ring

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.

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On transfer of annihilator conditions of rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501992 ◽

2018 ◽

Vol 17 (10) ◽

pp. 1850199

Author(s):

Abdollah Alhevaz ◽

Ebrahim Hashemi ◽

Rasul Mohammadi

Keyword(s):

Power Series ◽

Commutative Ring ◽

Formal Power Series ◽

Noetherian Rings ◽

Power Series Ring ◽

Series Ring ◽

Formal Power Series Ring ◽

Base Ring ◽

Polynomial Formula ◽

Formal Power

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

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The Sufficient Conditions for M[[S,w]] to be T[[S,w]]-Noetherian R[[S,w]]-module

Al-Jabar Jurnal Pendidikan Matematika ◽

10.24042/ajpm.v10i2.5042 ◽

2019 ◽

Vol 10 (2) ◽

pp. 285-292

Author(s):

Ahmad Faisol ◽

Fitriani Fitriani

Keyword(s):

Power Series ◽

Sufficient Conditions ◽

Power Series Ring ◽

Generalized Power Series ◽

Series Ring ◽

Ordered Monoid ◽

Monoid Homomorphism

In this paper, we investigate the sufficient conditions for T[[S,w]] to be a multiplicative subset of skew generalized power series ring R[[S,w]], where R is a ring, T Í R a multiplicative set, (S,≤) a strictly ordered monoid, and w : S®End(R) a monoid homomorphism. Furthermore, we obtain sufficient conditions for skew generalized power series module M[[S,w]] to be a T[[S,w]]-Noetherian R[[S,w]]-module, where M is an R-module.

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LEFT APP-RINGS OF SKEW GENERALIZED POWER SERIES

Journal of Algebra and Its Applications ◽

10.1142/s0219498811005014 ◽

2011 ◽

Vol 10 (05) ◽

pp. 891-900 ◽

Cited By ~ 4

Author(s):

RENYU ZHAO

Keyword(s):

Power Series ◽

Chain Condition ◽

Polynomial Rings ◽

Group Rings ◽

Power Series Ring ◽

Generalized Power Series ◽

Series Ring ◽

Ordered Monoid ◽

Skew Polynomial Rings ◽

Monoid Homomorphism

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any a ∈ R. Let R be a ring, (S, ≤) be a commutative strictly ordered monoid and ω: S → End (R) be a monoid homomorphism. The skew generalized power series ring [[RS, ≤, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings and Malcev–Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring [[RS, ≤, ω]]. It is shown that if (S, ≤) is a commutative strictly totally ordered monoid, ω: S→ Aut (R) a monoid homomorphism and R a ring satisfying the descending chain condition on right annihilators, then [[RS, ≤, ω]] is left APP if and only if for any S-indexed subset A of R, the ideal lR(∑a ∈ A ∑s ∈ S Rωs (a)) is right s-unital.

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ROTA–BAXTER OPERATORS ON GENERALIZED POWER SERIES RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498809003515 ◽

2009 ◽

Vol 08 (04) ◽

pp. 557-564 ◽

Cited By ~ 3

Author(s):

LI GUO ◽

ZHONGKUI LIU

Keyword(s):

Power Series ◽

Laurent Series ◽

Quantum Field ◽

Power Series Ring ◽

Generalized Power Series ◽

Series Ring ◽

Ordered Monoid ◽

Theory Application ◽

Power Series Rings ◽

Special Case

An important instance of Rota–Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota–Baxter operator and how this is related to the ordered monoid.

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Nonnil-Noetherian Rings and Formal Power Series

Algebra Colloquium ◽

10.1142/s1005386720000292 ◽

2020 ◽

Vol 27 (03) ◽

pp. 361-368

Author(s):

Ali Benhissi

Keyword(s):

Power Series ◽

Commutative Ring ◽

Formal Power Series ◽

Quotient Ring ◽

Noetherian Rings ◽

Power Series Ring ◽

Series Ring ◽

Formal Power

Let A be a commutative ring with unit. We characterize when A is nonnil-Noetherian in terms of the quotient ring A/ Nil(A) and in terms of the power series ring A[[X]].

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On Nonnil-S-Noetherian Rings

Mathematics ◽

10.3390/math8091428 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1428 ◽

Cited By ~ 1

Author(s):

Min Jae Kwon ◽

Jung Wook Lim

Keyword(s):

Power Series ◽

Commutative Ring ◽

Polynomial Ring ◽

Noetherian Ring ◽

Type Theorem ◽

Noetherian Rings ◽

Ring Extension ◽

Power Series Ring ◽

Flat Extension ◽

Series Ring

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

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