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.

Rings in Which the Annihilator of an Ideal Is Pure

2015 ◽  
Vol 22 (spec01) ◽  
pp. 947-968 ◽  
Author(s):  
A. Majidinya ◽  
A. Moussavi ◽  
K. Paykan
Keyword(s):  
Triangular Matrix ◽  
Complete Characterization ◽  
Direct Products ◽  
Matrix Rings ◽  
Morita Invariant ◽  
Sheaf Representation ◽  
Baer Rings ◽  
Upper Triangular Matrix ◽  
Left Annihilator

A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

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.

scholarly journals Functional identities on upper triangular matrix rings

2020 ◽  
Vol 18 (1) ◽  
pp. 182-193
Author(s):  
He Yuan ◽  
Liangyun Chen
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Unital Ring ◽  
Functional Identities ◽  
Upper Triangular Matrix ◽  
Free Subset

Abstract Let R be a subset of a unital ring Q such that 0 ∈ R. Let us fix an element t ∈ Q. If R is a (t; d)-free subset of Q, then Tn(R) is a (t′; d)-free subset of Tn(Q), where t′ ∈ Tn(Q), $\begin{array}{} t_{ll}' \end{array} $ = t, l = 1, 2, …, n, for any n ∈ N.

scholarly journals On p.p.-rings which are reduced

2006 ◽  
Vol 2006 ◽  
pp. 1-5
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Upper Triangular Matrix

Denote the2×2upper triangular matrix rings overℤandℤpbyUTM2(ℤ)andUTM2(ℤp), respectively. We prove that if a ringRis a p.p.-ring, thenRis reduced if and only ifRdoes not contain any subrings isomorphic toUTM2(ℤ)orUTM2(ℤp). Other conditions for a p.p.-ring to be reduced are also given. Our results strengthen and extend the results of Fraser and Nicholson on r.p.p.-rings.

scholarly journals Derivations of upper triangular matrix rings

1993 ◽  
Vol 187 ◽  
pp. 263-267 ◽  
Author(s):  
Sǒnia P. Coelho ◽  
C. Polcino Milies
Keyword(s):  
Triangular Matrix ◽  
Matrix Rings ◽  
Upper Triangular Matrix
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