Gaussian properties of total rings of quotients

Bazzoni, Silvana; Glaz, Sarah

doi:10.1016/j.jalgebra.2007.01.004

Cited by 61 publications

(115 citation statements)

References 12 publications

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“…Glaz [30] showed that (1) implies (5) while Lucas [42] showed the reverse implication (5) implies (1). Bazzoni and Glaz [7] proved the equivalence of (1) and (4). In addition, Tsang showed that the prime ideals in a local Gaussian ring are totally ordered.…”

Section: Theorem 36 ([2351]) Let R Be An Integral Domain Then R Ismentioning

confidence: 91%

“…The first approach, taken by Glaz in [30] utilizes conditions which control zero divisors. The second approach, taken by Bazzoni and Glaz in [7] and by Boynton in [10], consists of imposing conditions on Q(R). We describe the former approach first, starting with a brief background on zero divisor controlling conditions.…”

Section: Conditions That Allow Reversal Of Implicationsmentioning

confidence: 99%

“…It should also be noted that if R is a domain, that is, Q(R) is a field, then all of the Prüfer conditions are equivalent. The first step of this approach was to determine if Q(R) inherits the Prüfer condition of R. Conditions 1-4 and 6 are treated in [7], while condition 5 is treated in [10]. Below is the complete picture:…”

Section: Is a Gaussian Ring And A Pf Ring (3) R Is A Gaussian Ring Amentioning

confidence: 99%

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Prüfer Conditions in Commutative Rings

Glaz

Schwarz

2011

Arab J Sci Eng

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This article explores several extensions of the Prüfer domain notion to rings with zero divisors. These extensions include semihereditary rings, rings with weak global dimension less than or equal to 1, arithmetical rings, Gaussian rings, locally Prüfer rings, strongly Prüfer rings, and Prüfer rings. The renewed interest in these properties, due to their connection to Kaplansky's Conjecture, has resulted in a large body of results shedding new light on the area. We survey the work done in this direction in the last 15 years, including results, examples and counterexamples, and a multitude of open problems.

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Section: Theorem 36 ([2351]) Let R Be An Integral Domain Then R Ismentioning

confidence: 91%

Section: Conditions That Allow Reversal Of Implicationsmentioning

confidence: 99%

Section: Is a Gaussian Ring And A Pf Ring (3) R Is A Gaussian Ring Amentioning

confidence: 99%

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Prüfer Conditions in Commutative Rings

Glaz

Schwarz

2011

Arab J Sci Eng

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“…There are many rings that are not domain, but still Gaussian. For more on Gaussian rings, one may refer to [1], [4], [2], and [6]. In the next section, we will define Gaussian algebras and discuss them.…”

Section: Introductionmentioning

confidence: 99%

On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture

Nasehpour¹

2016

Arch. Math. (Brno)

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On the Anderson-Badawi ω R[X] (I[X]) = ω R (I) conjectureArchivum Mathematicum, Vol. 52 (2016) ON THE ANDERSON-BADAWI ω R[X] (I[X]) = ω R (I) CONJECTURE Peyman NasehpourAbstract. Let R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper on n-absorbing ideals, define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x 1 . . . x n+1 ∈ I for x 1 , . . . , x n+1 ∈ R, then there are n of the x i 's whose product is in I and conjecture that ω R [X] (I[X]) = ω R (I) for any ideal I of an arbitrary ring R, where ω R (I) = min{n : I is an n-absorbing ideal of R}. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold:(1) The ring R is a Prüfer domain.(2) The ring R is a Gaussian ring such that its additive group is torsion-free.(3) The additive group of the ring R is torsion-free and I is a radical ideal of R. IntroductionLet R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper [3], define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x 1 . . . x n+1 ∈ I for x 1 , . . . , x n+1 ∈ R, then there are n of the x i 's whose product is in I. In the fourth section of their paper, they conjecture that ω R [X] (I[X]) = ω R (I) for any ideal I of an arbitrary ring R, where ω R (I) = min{n : I is an n-absorbing ideal of R}.Clearly a 1-absorbing ideal is just a prime ideal and it is a well-known result in commutative ring theory that I is a prime ideal of R iff I[X] is a prime ideal of R [X]. In [3, Theorem 4.15], it is also proved that I[X] is a 2-absorbing ideal of R[X] iff I is a 2-absorbing ideal of R.In this paper, we use content formula techniques to prove that their conjecture is true, i.e., ω R[X] (I[X]) = ω R (I) for an ideal I of R, if one of the following conditions hold:(1) The ring R is a Prüfer domain.2010 Mathematics Subject Classification: primary 13A15; secondary 13B02, 13B25, 13F05.

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“…• R contains a prime ideal L such that L R L is non-zero and T-nilpotent (a slight generalization of [2,Theorem 6.4] using [1, Theorems P and 6.3]).…”

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confidence: 99%

Finitistic weak dimension of commutative arithmetical rings

Couchot

2012

Arab. J. Math.

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It is proven that each commutative arithmetical ring R has a finitistic weak dimension ≤ 2. More precisely, this dimension is 0 if R is locally IF, 1 if R is locally semicoherent and not IF, and 2 in the other cases. Mathematics Subject Classification 13F05 · 13F30All rings in this paper are unitary and commutative. A ring R is said to be a chain ring if its lattice of ideals is totally ordered by inclusion, and R is called arithmetical if R P is a chain ring for each maximal ideal P. If M is an R-module, we denote by w. When R is an arithmetical ring, by [9], a paper by Osofsky, we know that w.gl.d(R) ≤ 1 if R is reduced, and w.gl.d(R) = ∞ otherwise. By using the small finitistic dimension, similar results are proved by Glaz and Bazzoni when R is a Gaussian ring satisfying one of the following two conditions:

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Gaussian properties of total rings of quotients

Cited by 61 publications

References 12 publications

Prüfer Conditions in Commutative Rings

Prüfer Conditions in Commutative Rings

On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture

Finitistic weak dimension of commutative arithmetical rings

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