On skew Hurwitz serieswise Armendariz rings

2014 ◽  
Vol 07 (03) ◽  
pp. 1450036 ◽  
Author(s):  
M. Ahmadi ◽  
A. Moussavi ◽  
V. Nourozi
Keyword(s):  
Series Ring ◽  
Skew Hurwitz Series Ring ◽  
Ring Endomorphism ◽  
Armendariz Rings

For a ring endomorphism α, we introduce and investigate skew Hurwitz serieswise Armendariz (or SHA) rings which are a generalization of α-rigid rings and determine the radicals of the skew Hurwitz series ring (HR, α), in terms of those of R. We prove that several properties transfer between R and the extensions, in case R is an SHA-ring. We also construct various types of nonreduced SHA-rings.

Skew Hurwitz serieswise quasi-armendariz rings

2020 ◽  
pp. 2150118
Author(s):  
Kamal Paykan ◽  
Abdolreza Tehranchi
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Series Ring ◽  
Skew Hurwitz Series Ring ◽  
Ring Endomorphism ◽  
Semiprime Rings ◽  
Armendariz Rings ◽  
Baer Rings ◽  
Upper Triangular Matrix ◽  
Series Type

For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.

Differential Inverse Power Series Rings with Quasi-Armendariz-like Condition

2018 ◽  
Vol 25 (04) ◽  
pp. 595-618 ◽  
Author(s):  
Kamal Paykan ◽  
Abasalt Bodaghi
Keyword(s):  
Power Series ◽  
Triangular Matrix ◽  
Inverse Power ◽  
Direct Sums ◽  
Matrix Rings ◽  
Series Ring ◽  
Semiprime Rings ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Series Type

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, [Formula: see text]-quasi-Armendariz), is introduced and studied. It is shown that the [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime [Formula: see text]-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

Quasi-Armendariz generalized power series rings

2016 ◽  
Vol 15 (05) ◽  
pp. 1650086 ◽  
Author(s):  
K. Paykan ◽  
A. Moussavi
Keyword(s):  
Power Series ◽  
Triangular Matrix ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Matrix Rings ◽  
Generalized Power Series ◽  
Series Ring ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Upper Triangular Matrix

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] 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. We initiate the study of the [Formula: see text]-quasi-Armendariz condition on [Formula: see text], a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The [Formula: see text] formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

Extensions of Generalized Armendariz Rings

2006 ◽  
Vol 13 (02) ◽  
pp. 253-266 ◽  
Author(s):  
Chan Yong Hong ◽  
Tai Keun Kwak ◽  
S. Tariq Rizvi
Keyword(s):  
Polynomial Ring ◽  
Skew Polynomial Ring ◽  
Strong Connection ◽  
Principal Ideals ◽  
Ring Endomorphism ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

A ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. Such rings have been extensively studied in literature. For a ring endomorphism α, we introduce the notion of α-Armendariz rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established, and connections of properties of an α-Armendariz ring R with those of the ring R[x; α] are investigated. In particular, among other results, we show that there is a strong connection of the Baer property and the p.p.-property (principal ideals are projective) of the two rings, respectively. Several known results follow as consequences of our results.

scholarly journals A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

2010 ◽  
Vol 81 (3) ◽  
pp. 361-397 ◽  
Author(s):  
GREG MARKS ◽  
RYSZARD MAZUREK ◽  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Power Series ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Special Cases ◽  
Armendariz Rings

AbstractLet 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. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

On Armendariz Rings of Inverse Skew Laurent Series Type

2015 ◽  
Vol 22 (spec01) ◽  
pp. 799-816 ◽  
Author(s):  
Mohammad Habibi
Keyword(s):  
Laurent Series ◽  
Sufficient Conditions ◽  
Series Ring ◽  
Armendariz Rings ◽  
Series Type

Let R be a ring equipped with an automorphism α and an α-derivation δ. We study Armendariz rings of inverse (α,δ)-skew Laurent series type ((α,δ)-𝙸𝙻𝙰 rings) as a generalization of the standard Armendariz condition from polynomials to inverse skew Laurent series. We resolve the structure of (α,δ)-𝙸𝙻𝙰 rings and obtain various necessary or sufficient conditions for a ring to be (α,δ)-𝙸𝙻𝙰, unifying and generalizing a number of known Armendariz-like conditions. Also, we study relations between the set of annihilators in R and the set of annihilators in the inverse skew Laurent series ring R((x-1;α,δ)). For an α-compatible (α,δ)-𝙸𝙻𝙰 ring R, we prove that several properties transfer between R and R((x-1;α,δ)). Moreover, the radical of R((x-1;α,δ)) is determined in an α-compatible (α,δ)-𝙸𝙻𝙰 ring R.

scholarly journals REVERSIBLE SKEW GENERALIZED POWER SERIES RINGS

2011 ◽  
Vol 84 (3) ◽  
pp. 455-457
Author(s):  
A. R. NASR-ISFAHANI
Keyword(s):  
Power Series ◽  
Semiprime Ring ◽  
Unified Approach ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Monoid Homomorphism ◽  
Armendariz Rings ◽  
Power Series Rings ◽  
Unique Product Monoid

AbstractIn this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].

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.

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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