Radicals of skew polynomial rings and skew Laurent polynomial rings

Hong, Chan Yong; Kim, Nam Kyun; Lee, Yang

doi:10.1016/j.jalgebra.2010.12.028

Cited by 13 publications

(3 citation statements)

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“…Equivalently, the elements ax l ∈ R[x; σ] are nilpotent for each integer l ≥ 1 (cf. [13], [24], [26], etc.). We recall that a subset S of a ring R is called σ-nilpotent if for any integer l ≥ 1, there exists a positive integer m = m(l), depending on l, such that Sσ l (S)σ 2l (S) · · · σ (m−1)l (S) = 0 (cf.…”

Section: Skew Power-serieswise Armendariz Ringsmentioning

confidence: 99%

“…We recall that a subset S of a ring R is called σ-nilpotent if for any integer l ≥ 1, there exists a positive integer m = m(l), depending on l, such that Sσ l (S)σ 2l (S) · · · σ (m−1)l (S) = 0 (cf. [13]). Theorem 1.6.…”

Section: Skew Power-serieswise Armendariz Ringsmentioning

confidence: 99%

“…Pearson and Stephenson [24] defined the σ-prime radical of a ring R, denoted by P σ (R), as the intersection of all strongly σ-prime ideals of R. Many authors have used the concept of σ-prime ideal to study of radicals of skew polynomial rings, see ( [7], [13], [19], [23], [24], [25], [26], etc.). A subset S of a ring R is called σ-nil if every element in S is σ-nilpotent.…”

Section: Example 22mentioning

confidence: 99%

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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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Section: Skew Power-serieswise Armendariz Ringsmentioning

confidence: 99%

Section: Skew Power-serieswise Armendariz Ringsmentioning

confidence: 99%