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.

Unified Extensions of Strongly Reversible Rings and Links with Other Classic Ring Theoretic Properties

2018 ◽  
Vol 85 (3-4) ◽  
pp. 434
Author(s):  
R. K. Sharma ◽  
Amit B. Singh
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Group Rings ◽  
Laurent Polynomials ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings

Let R be a ring, (M, ≤) a strictly ordered monoid and ω : M → <em>End</em>(R) a monoid homomorphism. The skew generalized power series ring R[[M; ω]] is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev Neumann rings and generalized power series rings. In this paper, we introduce concept of strongly (M, ω)-reversible ring (strongly reversible ring related to skew generalized power series ring R[[M, ω]]) which is a uni ed generalization of strongly reversible ring and study basic properties of strongly (M; ω)-reversible. The Nagata extension of strongly reversible is proved to be strongly reversible if R is Armendariz. Finally, it is proved that strongly reversible ring strictly lies between reduced and reversible ring in the expanded diagram given by Diesl et. al. [7].

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].

On generalized power series rings with some restrictions on zero-divisors

2018 ◽  
Vol 17 (03) ◽  
pp. 1850040
Author(s):  
E. Hashemi ◽  
M. Yazdanfar ◽  
A. Alhevaz
Keyword(s):  
Power Series ◽  
Complete Characterization ◽  
Polynomial Rings ◽  
Group Rings ◽  
Property A ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Zero Divisor Graph ◽  
Power Series Rings

Let [Formula: see text] be a ring and [Formula: see text] a strictly ordered monoid. The construction of generalized power series ring [Formula: see text] generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring [Formula: see text] and a m.a.n.u.p. monoid [Formula: see text], it is shown that (i) [Formula: see text] is power-serieswise [Formula: see text]-McCoy, (ii) [Formula: see text] have Property (A), (iii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip, (iv) [Formula: see text] is strongly AB if and only if [Formula: see text] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring [Formula: see text] and the graph-theoretical properties of its undirected zero divisor graph of [Formula: see text]. A complete characterization for the possible diameters [Formula: see text] is given exclusively in terms of the ideals of [Formula: see text]. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.

Weak Normalization of Power Series Rings

1995 ◽  
Vol 38 (4) ◽  
pp. 429-433 ◽  
Author(s):  
David E. Dobbs ◽  
Moshe Roitman
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Power Series Ring ◽  
Series Ring ◽  
Power Series Rings

AbstractIt is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.

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.

Graded Complexes Over Power Series Rings

1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts
Keyword(s):  
Power Series ◽  
Local Ring ◽  
Simple Structure ◽  
Free Resolution ◽  
Koszul Complex ◽  
Local Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Noetherian Local Ring ◽  
Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

On semilocal, Bézout and distributive generalized power series rings

2015 ◽  
Vol 25 (05) ◽  
pp. 725-744 ◽  
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).

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.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Refaat M. Salem ◽  
Mohamed A. Farahat ◽  
Hanan Abd-Elmalk
Keyword(s):  
Power Series ◽  
Associative Ring ◽  
Ore Extension ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Unitary Right

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.

SKEW POWER SERIES RINGS OF DERIVATION TYPE

2011 ◽  
Vol 10 (06) ◽  
pp. 1383-1399 ◽  
Author(s):  
JEFFREY BERGEN ◽  
PIOTR GRZESZCZUK
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Noncommutative Algebra ◽  
Series Ring ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation ◽  
Power Series Rings

In this paper, we contrast the structure of a noncommutative algebra R with that of the skew power series ring R[[y;d]]. Several of our main results examine when the rings R, Rd, and R[[y;d]] are prime or semiprime under the assumption that d is a locally nilpotent derivation.

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