EXTENSIONS OF α-REFLEXIVE RINGS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250013 ◽  
Author(s):  
Liang Zhao ◽  
Xiaosheng Zhu
Keyword(s):  
Polynomial Rings ◽  
Rigid Ring ◽  
Basic Properties ◽  
Armendariz Ring ◽  
The Ideal

We introduce the notion of an α-reflexive ring to extend the concept of a reflexive ring and that of an α-rigid ring. We first consider some basic properties of α-reflexive rings, including some examples needed in the process. We prove that a ring R is α-rigid if and only if R is a reduced α-reflexive ring with α a monomorphism. We next investigate the α-reflexivity of some kinds of polynomial rings. It is shown that if R is a reduced α-reflexive ring with α(1) = 1, then R[x]/(xn) is an α-reflexive ring, where (xn) is the ideal generated by xn. Moreover, for an Armendariz ring R, we prove that R is α-reflexive if and only if R[x] is α-reflexive if and only if R[x; x-1] is α-reflexive. As a sequence, some known results relating to reflexive rings are obtained as corollaries of these results.

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.

ON Π-NEAR-ARMENDARIZ RINGS

2009 ◽  
Vol 02 (01) ◽  
pp. 77-83
Author(s):  
Sh. Ghalandarzadeh ◽  
P. Malakooti Rad
Keyword(s):  
Basic Properties ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Armendariz Ring

In this note we introduce a concept, so-called π-Near-Armendariz ring, that is a generalization of both Armendariz rings and 2-primal rings. We first observe the basic properties of π-Near-Armendariz rings, constructing typical examples. We next extend the class of π-Near-Armendariz rings, through various ring extensions.

Reflexive rings and their extensions

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Liang Zhao ◽  
Xiaosheng Zhu ◽  
Qinqin Gu
Keyword(s):  
Quotient Ring ◽  
Weak Form ◽  
Polynomial Rings ◽  
Armendariz Ring

AbstractA right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if R[x] is reflexive if and only if R[x; x −1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reflexive.

scholarly journals Ideal convergence in locally solid Riesz spaces

Filomat ◽  
2014 ◽  
Vol 28 (4) ◽  
pp. 797-809 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Bounded Sequence ◽  
Riesz Space ◽  
Ideal Convergence ◽  
Riesz Spaces ◽  
Basic Properties ◽  
Locally Solid Riesz Space ◽  
Positive Integers ◽  
The Ideal

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the concepts of ideal ?-convergence, ideal ?-Cauchy and ideal ?-bounded sequence in locally solid Riesz space endowed with the topology ?. Some basic properties of these concepts has been investigated. We also examine the ideal ?-continuity of a mapping defined on locally solid Riesz space.

Border bases and kernels of homomorphisms and of derivations

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Janusz Zieliński
Keyword(s):  
Polynomial Ring ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
The Ideal

AbstractBorder bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.

scholarly journals NS-Cross Entropy Based MAGDM under Single Valued Neutrosophic Set Environment

2018 ◽  
Author(s):  
Surapati Pramanik ◽  
Shyamal Dalapati ◽  
Shariful Alam ◽  
F. Smarandache ◽  
Tapan Kumar Roy
Keyword(s):  
Decision Making ◽  
Group Decision Making ◽  
Group Decision ◽  
Cross Entropy ◽  
Neutrosophic Set ◽  
Entropy Measure ◽  
Numerical Example ◽  
Basic Properties ◽  
Solution Strategies ◽  
The Ideal

Single valued neutrosophic set has king power to express uncertainty characterized by indeterminacy, inconsistency and incompleteness. Most of the existing single valued neutrosophic cross entropy bears an asymmetrical behavior and produce an undefined phenomenon in some situations. In order to deal with these disadvantages, we propose a new cross entropy measure under single valued neutrosophic set (SVNS) environment namely SN- cross entropy and prove its basic properties. Also we define weighted SN-cross entropy measure and investigate its basic properties. We develop a new multi attribute group decision making (MAGDM) strategy for ranking of the alternatives based on the proposed weighted SN-cross entropy measure between each alternative and the ideal alternative. Finally, a numerical example of MAGDM problem of investment potential is solved to show the validity and efficiency of proposed decision making strategy. We also present comparative anslysis of the obtained result with the results obtained form the existing solution strategies in the solution.

scholarly journals AN AMALGAMATED DUPLICATION OF A RING ALONG AN IDEAL: THE BASIC PROPERTIES

2007 ◽  
Vol 06 (03) ◽  
pp. 443-459 ◽  
Author(s):  
MARCO D'ANNA ◽  
MARCO FONTANA
Keyword(s):  
Topological Structure ◽  
General Construction ◽  
Prime Spectrum ◽  
Basic Properties ◽  
Amalgamated Duplication ◽  
The Ideal

We introduce a new general construction, denoted by R ⋈ E, called the amalgamated duplication of a ring R along an R-module E, that we assume to be an ideal in some overring of R. (Note that, when E2 = 0, R ⋈ E coincides with the Nagata's idealization R ⋉ E.). After discussing the main properties of the amalgamated duplication R ⋈ E in relation with pullback-type constructions, we restrict our investigation to the study of R ⋈ E when E is an ideal of R. Special attention is devoted to the ideal-theoretic properties of R ⋈ E and to the topological structure of its prime spectrum.

Idempotent Ideals and Noetherian Polynomial Rings

1982 ◽  
Vol 25 (1) ◽  
pp. 48-53 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Power Series ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Power Series Rings ◽  
Descending Chain Condition ◽  
The Ideal

AbstractIf R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.

Generalizations of reversible and Armendariz rings

2016 ◽  
Vol 26 (05) ◽  
pp. 911-933
Author(s):  
Juncheol Han ◽  
Tai Keun Kwak ◽  
Chang Ik Lee ◽  
Yang Lee ◽  
Yeonsook Seo
Keyword(s):  
Basic Properties ◽  
Armendariz Rings ◽  
Armendariz Ring

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of [Formula: see text]-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing [Formula: see text]-reversibility and power-Armendariz condition.

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