Armendariz Rings and Semicommutative Rings

Huh, Chan; Lee, Yang; Smoktunowicz, Agata

doi:10.1081/agb-120013179

Cited by 197 publications

(99 citation statements)

References 5 publications

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“…Any α-rigid ring is α-skew Armendariz, but the concepts of α-skew Armendariz rings and strongly α-skew reversible rings are independent of each other, by help of [11,Example 14] and [20,Example 3.2].…”

Section: Basic Properties Of Strongly α-Skew Reversible Ringsmentioning

confidence: 99%

On Commutativity of Skew Polynomials at Zero

Jin¹,

Kaynarca²,

Kwak³

et al. 2017

Bulletin of the Korean Mathematical Society

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Abstract. We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an α-rigid ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly α-skew reversible if the skew polynomial ring R[x; α] is reversible. We examine some characterizations and extensions of strongly α-skew reversible rings in relation with several ring theoretic properties which have roles in ring theory.

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Section: Basic Properties Of Strongly α-Skew Reversible Ringsmentioning

confidence: 99%

On Commutativity of Skew Polynomials at Zero

Jin¹,

Kaynarca²,

Kwak³

et al. 2017

Bulletin of the Korean Mathematical Society

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“…Then (a + b) s is an idempotent for some s ≥ 1 by the proof of [12,Proposition 16]. But (a + b) s is nonzero.…”

Section: Introductionmentioning

confidence: 92%

Semicommutative Property on Nilpotent Products

Kim¹,

Kwak²,

Lee³

2014

Journal of the Korean Mathematical Society

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Abstract. The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are σ-rigid δ-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where σ is a ring endomorphism and δ is a σ-derivation.

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“…We modify the construction and apply the computation in [15,Example]. Let Z 2 be the field of integers modulo 2 and A = Z 2 {a 0 , a 1 , a 2 , b 0 , b 1 , b 2 , c} be the free algebra of polynomials with zero constant terms in noncommuting indeterminates a 0 , a 1 , a 2 , b 0 , b 1 , b 2 , c over Z 2 .…”

Section: Example 22mentioning

confidence: 99%

Structure of Zero-Divisors in Skew Power Series Rings

Hong¹,

Kim²,

Lee³

2015

Journal of the Korean Mathematical Society

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Abstract. In this note we study the structures of power-serieswise Armendariz rings and IFP rings when they are skewed by ring endomorphisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the process. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.Throughout this paper every ring is an associative ring with identity. Let σ be an endomorphism of a ring R. We use R [[x; σ]] (resp. R[x; σ]) to denote the skew power series ring (resp. skew polynomial ring) with an indeterminate x over a ring R, subject to the relation xr = σ(r)x for r ∈ R. Note that σ(1) = 1 for any skew power series ring (skew polynomial ring) R [[x; σ]] (R[x; σ]), since 1x n = x n = x1x n−1 = σ(1)x n for any n ≥ 1 where 1 is the identity of R.Armendariz [2, Lemma 1] proved that whenever polynomialsover a reduced ring R satisfy f (x)g(x) = 0, then a i b j = 0 for all i, j. Rege and Chhawchharia [27] called such a ring (not necessarily reduced) Armendariz. The Armendariz condition, and various derivatives described below, have been studied by numerous authors. According to Kim et al. [16], we say that a ring R is power-serieswise Armendariz if a i b j = 0 for all i, j whenever power series However, we call the ring σ-skew power-serieswise Armendariz.According to Baser et al. [3, Definition 3.1], a ring R with an endomorphism σ is called σ-sps Armendariz if whenever power seriesIf we use the methods of [14, Theorem 1.8], we get the fact that σ-sps Armendariz rings are σ-skew power-serieswise Armendariz, but the converse is not true (see [14, Example 1.9]).Krempa [17] called an endomorphism σ of a ring R rigid if aσ(a) = 0 implies a = 0 for a ∈ R. We call a ring R σ-rigid if there exists a rigid endomorphism σ of R. Note that any rigid endomorphism of a ring is a monomorphism and σ-rigid rings are reduced rings by [12, Proposition 3]. Using this result, we can easily check that σ-rigid rings are σ-skew power-serieswise Armendariz.A ring R is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a, b ∈ R [4]. Narbonne In this note we will observe the previously mentioned conditions when they are equipped with an endomorphism (automorphism), and we study relationships between them. Skew power-serieswise Armendariz ringsIn this section, we study skew power-serieswise Armendariz rings which extend power-serieswise Armendariz rings and skew Armendariz rings, etc. We first note that σ-rigid rings are reduced rings by [12, Proposition 3]. Using this result, we can easily check that σ-rigid rings are σ-skew power-serieswise Armendariz. Moreover, we have the following result which gives a very short proof to Matczuk's result [21, Theorem A] and also contains a result [3, Theorem 3.3(1)]. Theorem 1.1. Let σ be an endomorphism of a ring R. Then the followi...

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Armendariz Rings and Semicommutative Rings

Cited by 197 publications

References 5 publications

On Commutativity of Skew Polynomials at Zero

On Commutativity of Skew Polynomials at Zero

Semicommutative Property on Nilpotent Products

Structure of Zero-Divisors in Skew Power Series Rings

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