Polynomial and power series ring extensions from sequences

2020 ◽  
pp. 2250048
Author(s):  
Gyu Whan Chang ◽  
Phan Thanh Toan
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Trivial Case ◽  
Addition Formula ◽  
Power Series Ring ◽  
Series Ring ◽  
Ring Extensions ◽  
General Sequences ◽  
Divisibility Properties ◽  
Positive Integers

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.

scholarly journals PF-rings of generalised power series

1998 ◽  
Vol 57 (3) ◽  
pp. 427-432 ◽  
Author(s):  
Zhongkui Liu
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Ordered Monoid

Let R be a commutative ring and (S, ≤) a strictly ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. We show that the generalised power series ring [[RS ≤]] is a PF-ring if and only if R is a PF-ring.

McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350083 ◽  
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.

On transfer of annihilator conditions of rings

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].

Nonnil-Noetherian Rings and Formal Power Series

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]].

scholarly journals On Nonnil-S-Noetherian Rings

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1428 ◽  
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.

scholarly journals Some strange behaviors of the power series ring R[[X]]

2018 ◽  
Vol 20 ◽  
pp. 01002
Author(s):  
Toan Phan Thanh
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Power Series Ring ◽  
Series Ring

Let R be a commutative ring with identity. Let R[X] and R[[X]] be the polynomial ring and the power series ring respectively over R. Being the completion of R[X] (under the X-adic topology), R[[X]] does not always share the same property with R[X]. In this paper, we present some known strange behaviors of R[[X]] compared to those of R[X].

scholarly journals A note on power invariant rings

1981 ◽  
Vol 4 (3) ◽  
pp. 485-491
Author(s):  
Joong Ho Kim
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Commutative Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Invariant Rings ◽  
Power Invariant

LetRbe a commutative ring with identity andR((n))=R[[X1,…,Xn]]the power series ring innindependent indeterminatesX1,…,XnoverR.Ris called power invariant if wheneverSis a ring such thatR[[X1]]≅S[[X1]], thenR≅S.Ris said to be forever-power-invariant ifSis a ring andnis any positive integer such thatR((n))≅S((n))thenR≅SLetIC(R)denote the set of alla∈Rsuch that there isR- homomorphismσ:R[[X]]→Rwithσ(X)=a. ThenIC(R)is an ideal ofR. It is shown that ifIC(R)is nil,Ris forever-power-invariant

scholarly journals Two properties of the power series ring

1988 ◽  
Vol 11 (1) ◽  
pp. 9-13 ◽  
Author(s):  
H. Al-Ezeh
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Pp Ring

For a commutative ring with unity,A, it is proved that the power series ringA〚X〛is a PF-ring if and only if for any two countable subsetsSandTofAsuch thatS⫅annA(T), there existsc∈annA(T)such thatbc=bfor allb∈S. Also it is proved that a power series ringA〚X〛is a PP-ring if and only ifAis a PP-ring in which every increasing chain of idempotents inAhas a supremum which is an idempotent.

scholarly journals When is a power series ring n-root closed?

1997 ◽  
Vol 114 (2) ◽  
pp. 111-131 ◽  
Author(s):  
David F. Anderson ◽  
David E. Dobbs ◽  
Moshe Roitman
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Series Ring
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