Absolute E-rings

Göbel, Rüdiger; Herden, Daniel; Shelah, Saharon

doi:10.1016/j.aim.2010.06.019

Cited by 4 publications

(12 citation statements)

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“…Then we must extend the notion of an E-ring and introduce E-morphisms; see Definition 3.3. It will follow that for suitable localizations R ⊆ R ′ of rings the inclusion is an E-morphism; see Theorem 3.7, which extends the main result (Theorem 3.1 from [22]). Certain absolute R-modules M from [18] have a desired prescribed R-endomorphism algebra A, and moreover M ⊗ R ′ becomes a free R ′ -module.…”

Section: Introductionmentioning

confidence: 73%

“…Our proof will be based on two previous results about absolute E-rings [22] and the existence of modules with absolute endomorphism rings in [18]. An analysis of the algebraic part of the rings R in [22] will show that for any R-modules M follows Hom Z (R/Z, M) = 0 which is essential for E-modules.…”

Section: Introductionmentioning

confidence: 98%

“…We will require that R has a family Π 4 of four distinct primes p such that R is p-reduced, which is an immediate consequence of the construction of the absolute E-rings in [22]. Then we must extend the notion of an E-ring and introduce E-morphisms; see Definition 3.3.…”

Section: Introductionmentioning

confidence: 99%

“…An analysis of the algebraic part of the rings R in [22] will show that for any R-modules M follows Hom Z (R/Z, M) = 0 which is essential for E-modules.…”

Section: Introductionmentioning

confidence: 99%

“…Throughout we will assume that • (I) R is a commutative ring with a family Π 4 of (at least) four distinct primes, and any R-algebra A under investigation will be p-reduced for all p ∈ Π 4 . • (II) We also assume for the construction of absolute E-rings R as in [22] that R is p-reduced for an infinite family of primes.…”

Section: Introductionmentioning

confidence: 99%

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Absolute E-modules

Göbel

2011

Journal of Pure and Applied Algebra

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Communicated by C.A. Weibel MSC: Primary: 13C05; 13C10; 13C13; 20K20; 20K25; 20K30 Secondary: 03E05; 03E35 a b s t r a c t Let R be a ring with 1 and M a right R-module. Then M is called E-module if Hom Z (R, M) = Hom R (R, M). Thus all homomorphisms between the abelian groups R Z and M Z turn out to be R-homogeneous. If M = R, then R is called an E-ring. It is clear from the definition, that the existence of E-modules requires that R be an E-ring and simple examples of E-rings are all subrings of Q. These structures are studied for more than thirty years in relation to questions concerning properties of modules, non-commutative groups and in the context of algebraic topology, see Casacuberta et al. (2010) [3] and Göbel and Trlifaj (2006) [24]. Constructions of large classes of E-rings were given in Faticoni (1987) [13] and Dugas et al. (1987) [9]; in a subsequent paper Dugas (1991) [6] showed the existence of large E-modules over R. More recently (initiated by Eklof and Shelah (1999) [11]) absolute algebraic structure came into the focus of studies see also Eklof and Mekler (2002) [10, p. 487 ff.]. Their advantage is, that they are robust under changes of the universe as explained in the introduction. We want to show the existence of absolute E-modules. On one hand, from earlier work on absolute E-rings Herden and Shelah (2009) [25] it follows that they must be smaller than κ(ω), which is the first ω-Erdős cardinal. Like measurable cardinals, κ(ω) is a large cardinal and may not exist. On the other hand, for each infinite cardinal λ an absolute E-ring of cardinality λ < κ(ω) exists, as shown in Göbel et al. (2010) [22]. Given such an absolute E-ring R, then we want to construct an absolute Emodule M over R for any cardinality |R| ≤ |M| < κ(ω) and study the decompositions of M.

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Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…An analysis of the algebraic part of the rings R in [22] will show that for any R-modules M follows Hom Z (R/Z, M) = 0 which is essential for E-modules.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Absolute E-modules

Göbel

2011

Journal of Pure and Applied Algebra

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Rings whose multiplicative endomorphisms are power functions

Oman

2012

Semigroup Forum

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Строение И Колебательные Спектры Димерного Комплексного Фторида Галлия (III) С Катионом Тетраметиламмония

Войт¹,

Давидович²,

Удовенко³

et al. 2019

Журнал Технической Физики

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Определена кристаллическая структура димерного комплексного фторида галлия(III) [N(CH_3)_4]2[Ga2F8(H2O)2], составленная из тетраэдрических катионов [N(CH_3)_4]+ и димерных комплексных анионов [Ga2F8(H2O)_2]2-, образованных объединением двух слегка искаженных октаэдрических групп GaF5(H2O) общим ребром F-F. Водородными связями O-H·sF димерные комплексные анионы [Ga2F8(H2O)_2]2- объединяются в полимерные цепи, между которыми расположены катионы тетраметиламмония. На основании квантово-химических расчетов выполнено отнесение полос в колебательных спектрах синтезированного соединения. В спектрах идентифицированы полосы, относящиеся к участвующим в водородных связях колебаниям молекул H2O, аниона и катиона. Ключевые слова: галлий(III), комплексный фторид, кристаллическая структура, тетраметиламмоний, димер, колебательная спектроскопия.

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Absolute E-rings

Cited by 4 publications

References 20 publications

Absolute E-modules

Absolute E-modules

Rings whose multiplicative endomorphisms are power functions

Строение И Колебательные Спектры Димерного Комплексного Фторида Галлия (III) С Катионом Тетраметиламмония

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