Essentially closed radical classes

Loi, Nguyen Van

doi:10.1017/s1446788700024812

Cited by 8 publications

(5 citation statements)

References 5 publications

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“…Then J ∼ = J/J ∩ I ∼ = (J + I)/I A/I ∈ N , so J = 0 (as N (A) = 0). Thus A is an essential extension of I, so A ∈ V [10].…”

Section: Proof (I) Ifmentioning

confidence: 99%

A Note on Mal'tsev-Neumann Products of Radical Classes

Gardner¹

2018

International Electronic Journal of Algebra

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A radical class R of rings is elementary if it contains precisely those rings whose singly generated subrings are in R. Many examples of elementary radical classes are presented, and all those which are either contained in the Jacobson radical class or disjoint from it are described. There is a discussion of Mal'tsev products of radical classes in general, in which it is shown, among other things, that a product of elementary radical classes need not be a radical class, and if it is, it need not be elementary.

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“…Then J ∼ = J/J ∩ I ∼ = (J + I)/I A/I ∈ N , so J = 0 (as N (A) = 0). Thus A is an essential extension of I, so A ∈ V [10].…”

Section: Proof (I) Ifmentioning

confidence: 99%

A Note on Mal'tsev-Neumann Products of Radical Classes

Gardner¹

2018

International Electronic Journal of Algebra

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“…See [33], pp. 38,48,30,55,57. Only in the case of associative rings is it known that innitely steps may be necessary (Heinicke [45]; see also [38], p.64).…”

Section: The Groupmentioning

confidence: 99%

“…It was shown by Loi [57] that every essentially closed radical class of associative rings is closed under products and is strongly hereditary and thus is a variety and a semi-simple class. The non-trivial semi-simple radical classes dene upper radical classes which are special and hence hereditary so these radical semi-simple classes are essentially closed.…”

Section: Essential Extensionsmentioning

confidence: 99%

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Radical Classes Closed Under Products

Gardner

2013

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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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.

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“…In 1970, Stewart [10] characterised semisimple radical classes in terms of subdirect sums of a finite set of finite fields. In 1983, Loi [7] showed that a radical class is semisimple if and only if it is closed under essential extensions (also see Gardner [6]). The last two results naturally lead one to consider the classification of the essential covers of radicals in terms of semisimplicity.…”

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confidence: 99%

Essential covers and complements of radicals

Birkenmeier

Wiegandt²

1996

Bull. Austral. Math. Soc.

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We show that a radical has a semisimple essential cover if and only if it is hereditary and has a complement in the lattice of hereditary radicals. In 1971 Snider gave a full description of supernilpotent radicals which have a complement. Recently Beidar, Fong, Ke, and Shum have determined radicals with semisimple essential covers. Using their results, we are able to provide a lower radical representation of complemented subidempotent radicals. This completes Snider's description of hereditary complemented radicals.In the context of radical theory the usefulness of the essential cover operator £ has been known from Armendariz [2] and Rjabuhin [8], who showed that a semisimple class is closed under essential extension if and only if the corresponding radical class is hereditary. In 1970, Stewart [10] characterised semisimple radical classes in terms of subdirect sums of a finite set of finite fields. In 1983, Loi [7] showed that a radical class is semisimple if and only if it is closed under essential extensions (also see Gardner [6]). The last two results naturally lead one to consider the classification of the essential covers of radicals in terms of semisimplicity. In 1994, Birkenmeier [4] showed that the essential cover £p of a supernilpotent radical p is nearly a semisimple class: it is hereditary, closed under extensions, finite subdirect sums, arbitrary direct sums and products. Hence Ep only lacks the requirement of being closed under arbitrary subdirect sums to become a semisimple class. Thus the question arises in [4]: which supernilpotent classes have semisimple essential covers? Imposing this seemingly mild extra condition on the essential cover Sp has turned out to be very restrictive: none of the classical radicals have semisimple essential covers [5]. Recently, Beidar, Fong, Ke, and Shum [3] have fully described radicals having semisimple essential covers. Their description is reminiscent of Stewart's characterisation of radical semisimple classes [10].Working on the same problem we found an alternative solution: radicals whose essential covers are semisimple classes, are exactly the hereditary radicals which have

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Essentially closed radical classes

Cited by 8 publications

References 5 publications

A Note on Mal'tsev-Neumann Products of Radical Classes

A Note on Mal'tsev-Neumann Products of Radical Classes

Radical Classes Closed Under Products

Essential covers and complements of radicals

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