On Commutativity of Skew Polynomials at Zero

Jin, Hai-lan; Kaynarca, Fatma; Kwak, Tai Keun; Lee, Yang

doi:10.4134/bkms.b150623

Cited by 3 publications

(4 citation statements)

References 16 publications

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“…In Theorem 3.3, we prove that if the classical left quotient ring Q(R) of R exists, then R is σ-symmetric if and only if Q(R) is strongly σ-symmetric. In Proposition 3.4, we obtain the results proved in [2, Proposition 3.6] and [8,Proposition 3.8] without the condition 'σ(u) = u' for any central regular element u. Hence, we get a direct generalization of [13,Lemma 3.2] without any restriction on the endomorphism σ.…”

Section: Abeliansupporting

confidence: 60%

“…But σ-compatible rings need not be strongly σ-symmetric by [8,Example 2.11]. In the following theorem, we show the relation between σ-compatible rings and strongly σ-symmetric rings.…”

Section: Corollary 28 ([13 Proposition 34]) Let R Be An Armendariz Ri...mentioning

confidence: 87%

“…The following lemma is clear by [8,Lemma 2.3], since any strongly σ-symmetric ring is strongly σ-skew reversible. For the sake of completeness, we include the statements.…”

Section: Corollary 28 ([13 Proposition 34]) Let R Be An Armendariz Ri...mentioning

confidence: 99%

“…Another approach to generalize reversible and IFP properties is obtained by considering the properties on Ore extensions of endomorphism type. Following [8]…”

Section: Introductionmentioning

confidence: 99%

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On symmetric property of skew polynomial rings

Kaynarca¹,

Akcin²

2018

Preprint

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Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly σ-symmetric if the skew polynomial ring R[x; σ] is symmetric. We consider some properties and extensions of strongly σ-symmetric rings. Then we show the relationship between strongly σ-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly σ-symmetric rings. Moreover, we prove that if R is a σ-rigid ring, then R[x]/(x n ) is a strongly σ-symmetric ring, where σ is an endomorphism of R, (x n ) is the ideal generated by x n and n is a positive integer; and that if the classical left quotient ring Q(R) of R exists, then R is σ-symmetric if and only if Q(R) is strongly σ-symmetric.

show abstract

Section: Abeliansupporting

confidence: 60%

“…But σ-compatible rings need not be strongly σ-symmetric by [8,Example 2.11]. In the following theorem, we show the relation between σ-compatible rings and strongly σ-symmetric rings.…”

Section: Corollary 28 ([13 Proposition 34]) Let R Be An Armendariz Ri...mentioning

confidence: 87%

“…The following lemma is clear by [8,Lemma 2.3], since any strongly σ-symmetric ring is strongly σ-skew reversible. For the sake of completeness, we include the statements.…”

Section: Corollary 28 ([13 Proposition 34]) Let R Be An Armendariz Ri...mentioning

confidence: 99%

“…Another approach to generalize reversible and IFP properties is obtained by considering the properties on Ore extensions of endomorphism type. Following [8]…”

Section: Introductionmentioning

confidence: 99%

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On symmetric property of skew polynomial rings

Kaynarca¹,

Akcin²

2018

Preprint

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Extensions of i-reversible rings

et al. 2023

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A ring [Formula: see text] is said to be i-reversible if for every [Formula: see text] [Formula: see text][Formula: see text], [Formula: see text] is a nonzero idempotent implies [Formula: see text] is an idempotent. It is known that the rings [Formula: see text] and [Formula: see text] (the ring of all upper triangular matrices over [Formula: see text]) are not i-reversible for [Formula: see text]. In this paper, we provide a nontrivial i-reversible subring of [Formula: see text] when [Formula: see text] and [Formula: see text] has only trivial idempotents. We further provide a maximal i-reversible subring of [Formula: see text] for each [Formula: see text], if [Formula: see text] is a field. We then give conditions for i-reversibility of trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings.

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Reversibility of skew Hurwitz series rings

Kaynarca

Yildirim

2020

Hacettepe Journal of Mathematics and Statistics

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We study the reversibility of skew Hurwitz series at zero as a generalization of an αrigid ring, introducing the concept of skew Hurwitz reversibility. A ring R is called skew Hurwitz reversible (SH-reversible, for short), if the skew Hurwitz series ring (HR, α) is reversible i.e. whenever skew Hurwitz series f, g ∈ (HR, α) satisfy f g = 0, then gf = 0. We examine some characterizations and extensions of SH-reversible rings in relation with several ring theoretic properties which have roles in ring theory.

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On Commutativity of Skew Polynomials at Zero

Cited by 3 publications

References 16 publications

On symmetric property of skew polynomial rings

On symmetric property of skew polynomial rings

Extensions of i-reversible rings

Reversibility of skew Hurwitz series rings

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