Semi-simple radical classes of algebras and attainability of identities

Analele Universitatii "Ovidius" Constanta - Seria Matematica

2013

Self 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.

Section: Essential Extensionsmentioning

confidence: 99%

Radical Classes Closed Under Products

Analele Universitatii "Ovidius" Constanta - Seria Matematica

2013

Self Cite

“…In the case of universal classes which are varieties of rings and related structures, a fair amount is now known about the (sub) varieties which are radical or semi-simple classes (see, for example, [9]). We next consider these questions for idempotent algebras.…”

Section: Varietiesmentioning

confidence: 99%

“…Our examples discussed so far provide similar counterexamples. In varieties of ring and group-based structures [9] and in congruence-permutable varieties of idempotent algebras (3.5) subvarieties with attainable identities are SSR classes. The case of semilattices shows that the converse is false for idempotent algebras in general.…”

Section: Cb)(bc) = (Cb)(bc)(cb)mentioning

confidence: 99%

Radical decompositions of idempotent algebras

1984

J Aust Math Soc A

A variant of Kurosh-Amitsur radical theory is developed for algebras with a collection of (finitary) operations w, all of which are idempotent, that is satisfy the condition u (x, x,...,x) = x. In such algebras, all classes of any congruence are subalgebras. In place of a largest normal radical subobject, a largest congruence with radical congruence classes is considered. In congruence-permutable varieties the parallels with conventional radical theory are most striking.

“…(This terminology is due to Tamura (1966).) A semi-simple radical class is the same thing as a variety with attainable identities (Gardner (1975), Theorem 1.5). THEOREM 3.2.…”

Section: Semi-simple Radical Classesmentioning

confidence: 99%

Radicals related to the Brown-McCoy radical in some varieties of algebras

1979

J. Aust. Math. Soc.

The Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.