Primeness, semiprimeness and prime radical of ore extensions

Lam, T. Y.; Leroy, André; Matczuk, Jerzy

doi:10.1080/00927879708826000

Cited by 58 publications

(18 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notation [22]. Let δ be an α-derivation of R. For integers j i 0, let us write f j i for the sum all "words" in α and δ in which there are i factors of α and j − i factors of δ.…”

Section: Skew Polynomial and Power Series Rings Over Quasi-baer And Lmentioning

confidence: 99%

Polynomial extensions of quasi-Baer rings

2005

View full text Add to dashboard Cite

For a ring endomorphism α and an α-derivation δ, we introduce α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].

show abstract

“…Notation [22]. Let δ be an α-derivation of R. For integers j i 0, let us write f j i for the sum all "words" in α and δ in which there are i factors of α and j − i factors of δ.…”

Section: Skew Polynomial and Power Series Rings Over Quasi-baer And Lmentioning

confidence: 99%

Polynomial extensions of quasi-Baer rings

2005

View full text Add to dashboard Cite

show abstract

“…This shows that a is not strongly θ-nilpotent. □ [3] are different from ours. In Section 3 we will prove that rad(R;…”

Section: Corollary 14 R[x; θ] Is Semiprime If and Only If R Is θ-Sementioning

confidence: 70%

Characterizations of Elements in Prime Radicals of Skew Polynomial Rings and Skew Laurent Polynomial Rings

Cheon¹,

Kim²,

Lee³

et al. 2011

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…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%

Structure of Zero-Divisors in Skew Power Series Rings

Hong¹,

Kim²,

Lee³

2015

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

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...

show abstract

Primeness, semiprimeness and prime radical of ore extensions

Cited by 58 publications

References 11 publications

Polynomial extensions of quasi-Baer rings

Polynomial extensions of quasi-Baer rings

Characterizations of Elements in Prime Radicals of Skew Polynomial Rings and Skew Laurent Polynomial Rings

Structure of Zero-Divisors in Skew Power Series Rings

Contact Info

Product

Resources

About