McCoy rings and zero-divisors

Camillo, Victor; Nielsen, Pace P.

doi:10.1016/j.jpaa.2007.06.010

Cited by 80 publications

(40 citation statements)

References 15 publications

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“…Now let R be an IFP ring, and f (x), g(x) ∈ R[x] be nonzero polynomials satisfying f (x)g(x) = 0. In this situation, Camillo and Nielsen showed that if r R [x] (f (x)) ∩ R = 0 then deg f (x) > 2 in [7,Theorem 5.5]. We here elaborate upon this theorem by finding nonzero elements in R contained in the right annihilator of f (x) when the degree of f (x) is ≤ 2.…”

Section: Annihilators Of Polynomials On Ifp Ringsmentioning

confidence: 95%

An Elaboration of Annihilators of Polynomials

Cheon¹,

Kim²,

Kim³

et al. 2017

Bulletin of the Korean Mathematical Society

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Abstract. In this note we elaborate first on well-known theorems for annihilators of polynomials over IFP rings by investigating the concrete shapes of nonzero constant annihilators. We consider next a generalization of IFP which preserves Abelian property, in relation with annihilators of polynomials, observing the basic structure of rings satisfying such condition. Annihilators of polynomials on IFP ringsIFP and Abelian ring property have important roles in noncommutative ring theory and module theory. We continue in this section the studies of Nielsen [22] and Shin [25], being concerned with the constant annihilators of polynomials, and introduce a generalization of IFP which preserves Abelian property.Throughout this note every ring is an associative ring with identity unless otherwise stated. Given a ring R, let N (R), N * (R), and J(R) denote the set of all nilpotent elements, the prime radical, and the Jacobson radical in R, respectively. The polynomial (resp., power series) ring with an indeterminate x over R is denoted by R[x] (resp., R[[x]]). The right annihilator of S in R is denoted by r R (S), and by r R (a) when S = {a}. The degree of a polynomial f (x) is denoted by deg f (x). The n by n full (resp. upper triangular) matrix ring over R is denoted by Mat n (R) (resp. U n (R)), and denote by e ij the matrix with (i, j)-entry 1 and elsewhere zero. Z denotes the ring of integers, and Z n denotes the ring of integers modulo n.A ring R (possibly without identity) is called reduced if N (R) = 0. A wellknown property that unifies the commutativity and the reduced condition is the insertion-of-factors-property. Due to Bell [4], a ring R (possibly without identity) is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a, b ∈ R. Narbonne [21] and Shin [25] used the terms semicommutative and SI for the IFP, respectively. Commutative rings are clearly IFP, and any reduced ring is IFP by a simple computation.

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Section: Annihilators Of Polynomials On Ifp Ringsmentioning

confidence: 95%

An Elaboration of Annihilators of Polynomials

Cheon¹,

Kim²,

Kim³

et al. 2017

Bulletin of the Korean Mathematical Society

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“…We need the following result which is a generalization of [8,Theorem 8.2] to the more general setting: Theorem 2. 27.…”

Section: Lemma 25 ([19 Theorem 44])mentioning

confidence: 99%

“…We apply the method of Camillo and Nielsen in the proof of [8,Theorem 8.2]. For every γ ∈ R * M we let I γ denote the right ideal generated by the coefficients of γ.…”

Section: Lemma 25 ([19 Theorem 44])mentioning

confidence: 99%

On Annihilations of Ideals in Skew Monoid Rings

Mohammadi¹,

Moussavi²,

Zahiri³

2016

Journal of the Korean Mathematical Society

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Abstract. According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and σ : M → End(R) a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and σ : M → End(R) is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

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“…In 2006, Nielsen [15] gave an example of semi-commutative ring which is not right McCoy. The concept of a linearly McCoy ring, which properly generalizes McCoy rings and semi-commutative rings, was introduced by Camillo and Nielsen [4] . Related results on McCoy conditions can be found in [4,5,10,12,15,16,17], etc.…”

Section: Introductionmentioning

confidence: 99%

“…The concept of a linearly McCoy ring, which properly generalizes McCoy rings and semi-commutative rings, was introduced by Camillo and Nielsen [4] . Related results on McCoy conditions can be found in [4,5,10,12,15,16,17], etc. Recently, the McCoy and the Armendariz conditions were extended to their module versions (see [3,6]).…”

Section: Introductionmentioning

confidence: 99%

Extensions of linearly McCoy rings

Cui¹,

Chen²

2013

Bulletin of the Korean Mathematical Society

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Abstract. A ring R is called linearly McCoy if whenever linear polynomials f (x), g(x) ∈ R[x]\{0} satisfy f (x)g(x) = 0, there exist nonzero elements r, s ∈ R such that f (x)r = sg(x) = 0. In this paper, extension properties of linearly McCoy rings are investigated. We prove that the polynomial ring over a linearly McCoy ring need not be linearly McCoy. It is shown that if there exists the classical right quotient ring Q of a ring R, then R is right linearly McCoy if and only if so is Q. Other basic extensions are also considered.

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McCoy rings and zero-divisors

Cited by 80 publications

References 15 publications

An Elaboration of Annihilators of Polynomials

An Elaboration of Annihilators of Polynomials

On Annihilations of Ideals in Skew Monoid Rings

Extensions of linearly McCoy rings

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