On Rings Having McCoy-Like Conditions

Alhevaz, Abdollah; Moussavi, A.; Habibi, M.

doi:10.1080/00927872.2010.548842

Cited by 21 publications

(3 citation statements)

References 38 publications

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“…For an endomorphism a and an a-derivation d of a ring R, Alhevaz et al [1] studied a generalization of the (a; d)skew Armendariz rings and the weak Armendariz rings which is called weak (a; d)-skew Armendariz rings. A ring R is called weak (a; d)-skew Armendariz ring, if whenever polynomials pðxÞ ¼ P m i¼0 s i x i and qðxÞ ¼ P n j¼0 t j x j 2 R½x; a; d satisfy pðxÞqðxÞ ¼ 0, then s i x i t j x j 2 nilðRÞ½x; a; d, for each i and j, where nil(R) denotes the set of all the nilpotent elements of R.…”

Section: Introductionmentioning

confidence: 99%

Weak ( $$\alpha ,\delta$$ α , δ )-skew Armendariz ideals

et al. 2018

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In this paper, we continue to study properties of skew polynomial ring R½x; a; d, where R is an associative ring equipped with an endomorphism a and an a-derivation d. We first introduce weak (a; d)-skew Armendariz ideals, a generalization of (a; d)-compatible ideals have the insertion of factors property (or simply, IFP). We also prove that if J is a weak (a; d)-skew Armendariz ideal of R, then J[x] is a weak (a; d)-skew Armendariz ideal of R[x]. Keywords (a Á d)-Skew Armendariz Á Weak Armendariz rings Á Weak (a Á d)-skew Mathematics Subject Classification 16S36 Á 16S50 Á 16S99 Hashemi [2], for an endomorphism a and an a-derivation d of a ring R, defined (a; d)-compatible ideals. An ideal J of R say that J is an a-compatible ideal of R if for each s; t 2 R; st 2 J () saðtÞ 2 J. Moreover, J is said to be a dcompatible ideal if for each s; t 2 R; st 2 J ¼) sdðtÞ 2 J. If J is both a-compatible and d-compatible, we say that J is an (a; d)-compatible ideal. In this article, first we introduce weak (a; d)-skew Armendariz ideals. Then, we show that if an (a; d)-compatible ideal J has the IFP, then J is weak (a; d)-skew Armendariz ideal; thus, weak (a; d)-skew Armendariz ideals are a generalization of (a; d)-compatible ideals which has the IFP. Also, for any positive integer n, we study relationship between ideals of R which are weak (a; d)-skew Armendariz with some ideals of the n-byn upper triangular matrices ring with constant main diagonal, the n-by-n upper triangular matrix ring over R and the ring R½x ðx n Þ , where ðx n Þ is the ideal generated by x n. Also we show that if J is an ideal of R, then J is weak (a; d)-skew Armendariz ideal if and only if J[x] is a weak (a; d)-skew Armendariz ideal of R[x].

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Section: Introductionmentioning

confidence: 99%

Weak ( $$\alpha ,\delta$$ α , δ )-skew Armendariz ideals

et al. 2018

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“…On the other hand, Alhevaz et al [1] defined a skew triangular matrix ring as follows. Let σ be an endomorphism of a ring R with σ(1) = 1.…”

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confidence: 99%

On extensions of matrix rings with skew Hochschild 2-cocycles

et al. 2016

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We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.Keywords Skew Hochschild extensions, matrix rings, skew triangular matrix rings, (uniquely) clean rings, symmetric rings MSC 16S50, 15A30, 15B33 Throughout this paper, R denotes an associative ring with identity. We use R[x] to denote the polynomial ring with an indeterminate x over R. For the factor ring R[x]/(x n+1 ), (x n+1 ) is the ideal of R[x] generated by x n+1 for n 1. Z (Z n ) denotes the ring of integers (modulo n).Following the literature, a ring is called reduced if it has no nonzero nilpotent elements. A ring R is called symmetric if abc = 0 implies acb = 0 for a, b, c ∈ R. A ring R is called reversible if ab = 0 implies ba = 0 for a, b ∈ R. A ring R is called semicommutative if ab = 0 implies aRb = 0 for a, b ∈ R. Clearly, reduced rings are symmetric, symmetric rings are reversible, and reversible rings are semicommutative, but the converse is not true in either case. A ring R is called

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“…Alhevaz et al [1] and Nasr-Isfahani [19] introduced a skew triangular matrix ring as a set of all upper triangular matrices with addition point-wise and new multiplication defined by…”

Section: Introductionmentioning

confidence: 99%

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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On Rings Having McCoy-Like Conditions

Cited by 21 publications

References 38 publications

Weak ( $$\alpha ,\delta$$ α , δ )-skew Armendariz ideals

Weak ( $$\alpha ,\delta$$ α , δ )-skew Armendariz ideals

On extensions of matrix rings with skew Hochschild 2-cocycles

Semicommutative Property on Nilpotent Products

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