On Weak Symmetric Rings

Ouyang, Lunqun; Chen, Huanyin

doi:10.1080/00927870902828702

Cited by 27 publications

(33 citation statements)

References 13 publications

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“…The nil-semicommutative property between R[x; σ, δ] and R is studied by Ouyang and Chen [21], when R is a (σ, δ)-compatible reversible ring. In this section, we continue to study for the nil-semicommutative property of R[x; σ, δ] and R[[x; σ]].…”

Section: Proof (1) This Is In [11 Corollary 23] (2) and (4)-(i) Fmentioning

confidence: 99%

“…Following [21], for integers i, j with 0 ≤ i ≤ j, let f j i ∈ End(R, +) be the map which is the sum of all possible words in σ, δ built with i letters σ and j − i letters δ. For example, f 0 0 = 1, f j j = σ i , f j 0 = δ j and f j j−1 = σ j−1 δ + σ j−2 δσ + · · · + δσ j−1 .…”

Section: Proof (1) This Is In [11 Corollary 23] (2) and (4)-(i) Fmentioning

confidence: 99%

“…(2) We apply the proof of [21,Lemma 2.10]. Suppose that p(x) ∈ N (R[x; σ, δ]) for p(x) = a 0 + a 1 x + · · · + a n x n ∈ R[x; σ, δ].…”

Section: Proof (1) This Is In [11 Corollary 23] (2) and (4)-(i) Fmentioning

confidence: 99%

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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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Section: Proof (1) This Is In [11 Corollary 23] (2) and (4)-(i) Fmentioning

confidence: 99%

Section: Proof (1) This Is In [11 Corollary 23] (2) and (4)-(i) Fmentioning

confidence: 99%

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Semicommutative Property on Nilpotent Products

Kim¹,

Kwak²,

Lee³

2014

Journal of the Korean Mathematical Society

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“…Reduced rings are symmetric by the results of Anderson and Camillo [1], but there are many nonreduced commutative (so symmetric) rings. As a generalization of symmetric rings, L. Ouyang introduced the notion of weak symmetric rings and showed that if R is an (α, δ)-compatible and reversible ring, then R is weak symmetric if and only if the Ore extension R[x; α, δ] is weak symmetric [16]. In the following, we investigate the weak symmetric property of the rings of generalized power series.…”

Section: And So Yf (S) ∈ Nil(r) For All Y ∈ Y and S ∈ S Thus F (S)mentioning

confidence: 99%

“…Any concept and notation not defined here can be founded in Ribenboim [17][18][19], Elliott and Ribenboim [6], and L. Ouyang [15][16].…”

Section: Introductionmentioning

confidence: 99%

Special Weak Properties of Generalized Power Series Rings

Ouyang¹

2012

Journal of the Korean Mathematical Society

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Abstract. Let R be a ring and nil(R) the set of all nilpotent elements of R. For a subset X of a ring R, we define N R (X) = {a ∈ R | xa ∈ nil(R) for all x ∈ X}, which is called a weak annihilator of X in R. A ring R is called weak zip provided that for any subset, and a ring R is called weak symmetric if abc ∈ nil(R) ⇒ acb ∈ nil(R) for all a, b, c ∈ R. It is shown that a generalized power series ring [[R S,≤ ]] is weak zip (resp. weak symmetric) if and only if R is weak zip (resp. weak symmetric) under some additional conditions. Also we describe all weak associated primes of the generalized power series ring [[R S,≤ ]] in terms of all weak associated primes of R in a very straightforward way.

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Nilpotent elements of skew Hurwitz series rings

Paykan

2016

Rend. Circ. Mat. Palermo, II. Ser

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On Weak Symmetric Rings

Cited by 27 publications

References 13 publications

Semicommutative Property on Nilpotent Products

Semicommutative Property on Nilpotent Products

Special Weak Properties of Generalized Power Series Rings

Nilpotent elements of skew Hurwitz series rings

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