Annihilator Conditions on Ore Extensions

2008 ◽  
Vol 15 (02) ◽  
pp. 293-302
Author(s):  
E. Hashemi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Polynomial Rings ◽  
Skew Polynomials ◽  
Baer Ring ◽  
Skew Polynomial Rings ◽  
Ore Extensions ◽  
Skew Polynomial ◽  
Armendariz Ring ◽  
Formal Power

In this paper, we extend the study of various annihilator conditions on the nearring of polynomials and the nearring of formal power series to skew polynomials and skew formal power series in which addition and substitution are used as operations. A result of Hong et al. on the skew polynomial rings and skew power series of an α-rigid Baer ring is extended to Baer conditions in the nearrings of skew polynomials and skew formal power series. Also, for an injective endomorphism α of a ring R, it is shown that R is α-rigid if and only if the nearring (R[x; α], +, ◦) is reduced, if and only if the nearring (R0[[x; α]], +, ◦) is reduced, if and only if R is a reduced and near Armendariz ring.

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250079 ◽  
Author(s):  
A. ALHEVAZ ◽  
A. MOUSSAVI
Keyword(s):  
Polynomial Rings ◽  
Trivial Extension ◽  
Skew Polynomial Ring ◽  
Matrix Rings ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Skew Polynomial ◽  
Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

On α-nilpotent elements and α-Armendariz rings

2015 ◽  
Vol 14 (05) ◽  
pp. 1550064
Author(s):  
Hong Kee Kim ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

scholarly journals A fresh look into monoid rings and formal power series rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050003
Author(s):  
Abolfazl Tarizadeh
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Universal Property ◽  
Polynomial Rings ◽  
Power Series Rings ◽  
Universal Properties ◽  
Ring Of Polynomials ◽  
Formal Power

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

ON WEAK ZIP SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 05 (03) ◽  
pp. 1250039
Author(s):  
R. Mohammadi ◽  
A. Moussavi ◽  
M. Zahiri
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
General Situation ◽  
Skew Polynomial Ring ◽  
Glasgow Math ◽  
Skew Polynomial Rings ◽  
Ore Extensions ◽  
Skew Polynomial ◽  
J 51

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil (R)[x; α, δ] = nil (R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.

Rota–Baxter operators on skew generalized power series rings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450048 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Power Series ◽  
Neumann Series ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Skew Polynomial

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) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.

Left principally quasi-Baer and left APP-rings of skew generalized power series

2014 ◽  
Vol 14 (03) ◽  
pp. 1550038 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Power Series ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Power Series Rings ◽  
Skew Polynomial ◽  
Classical Ring

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) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, and (skew) monoid rings. We characterize when a skew generalized power series ring R[[S, ω]] is left principally quasi-Baer and under various finiteness conditions on R we characterize when the ring R[[S, ω]] is left APP. As immediate corollaries we obtain characterizations for all aforementioned classical ring constructions to be left principally quasi-Baer or left APP. Such a general approach not only gives new results for several constructions simultaneously, but also serves the unification of already known results.

scholarly journals Centers of skew polynomial rings

2015 ◽  
Vol 97 (111) ◽  
pp. 181-186
Author(s):  
Waldo Arriagada ◽  
Hugo Ramírez
Keyword(s):  
Minimal Polynomial ◽  
Polynomial Rings ◽  
Skew Polynomials ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

We determine the center C(K[x;?]) of the ring of skew polynomials K[x;?], where K is a field and ? is a non-zero derivation over K. We prove that C(K[x;?]) = ker ?, if ? is transcendental over K. On the contrary, if ? is algebraic over K, then C(K[x;?])=(ker ?)[?(x)]. The term ?(x) is the minimal polynomial of ? over K.

Integer-valued skew polynomials

2020 ◽  
pp. 2150114
Author(s):  
Nicholas J. Werner
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

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