A characterization of ℵ1-free abelian groups and its application to the chase radical

Eda, Katsuya

doi:10.1007/bf02766167

Cited by 9 publications

(9 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…ℵ 1 : L({Q(p), Q(q)}) for distinct primes p, q [35]; the Chase radical (the upper radical class dened by the class of ℵ 1 -free abelian groups. This was proved by Eda [20] with the Continuum Hypothesis and by Pokutta and Strüngman [68] with ZFC.…”

Section: The Groupmentioning

confidence: 78%

Radical Classes Closed Under Products

Gardner

2013

Analele Universitatii "Ovidius" Constanta - Seria Matematica

View full text Add to dashboard Cite

This is a survey of what is known about Kurosh-Amitsur radical classes which are closed under direct products. Associative rings, groups, abelian groups, abelian -groups and modules are treated. We have attempted to account for all published results relevant to this topic. Many or most of these were not, as published, expressed in radical theoretic terms, but have consequences for radical theory which we point out. A fruitful source of results and examples is the notion of slenderness for abelian groups together with its several variants for other structures.We also present a few new results, including examples and a demonstration that e-varieties of regular rings are product closed radical classes of associative rings.

show abstract

Section: The Groupmentioning

confidence: 78%

Radical Classes Closed Under Products

Gardner

2013

Analele Universitatii "Ovidius" Constanta - Seria Matematica

View full text Add to dashboard Cite

show abstract

“…This norm R , if it exist, is always regular, see [6] and clearly, t = ℵ 0 . Eda [10] showed that satisfies ℵ 1 2 ℵ 0 . The value otherwise is quite arbitrary but depends on the underlying set theory.…”

Section: Introductionmentioning

confidence: 98%

How rigid are reduced products?

Göbel

Shelah

2005

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

For any cardinal let Z be the additive group of all integer-valued functionsIf are regular cardinals we analyze the question when Hom(Z , Z ) = 0 and obtain a complete answer under GCH and independence results in Section 8. These results and some extensions are applied to a problem on groups: Let the norm G of a group G be the smallest cardinal with Hom(Z , G) = 0-this is an infinite, regular cardinal (or ∞). As a consequence we characterize those cardinals which appear as norms of groups. This allows us to analyze another problem on radicals: The norm R of a radical R is the smallest cardinal for which there is a family {G i : i ∈ } of groups such that R does not commute with the product i∈ G i . Again these norms are infinite, regular cardinals and we show which cardinals appear as norms of radicals. The results extend earlier work (

show abstract

“…It is known that µ is always a regular uncountable cardinal and in [2] those cardinals were classified that can be realized as the norm λ = µ of some group radical µ. Moreover, a result due to Eda [4] shows that the norm of the Chase radical is less than or equal to 2 ℵ 0 , hence ν = ℵ 1 assuming the continuum hypothesis 2 ℵ 0 = ℵ 1 (CH).…”

Section: Introductionmentioning

confidence: 99%