Some Pathology for Radicals in Non-Associative Near-Rings

Stefan Veldsman

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

It is well-known that every semisimple class in the class of all associative rings or alternative rings is hereditary, this being an easy consequence of the Anderson-Divinsky-Suliński property (cf. [1]). Dropping the associativity, one get degenerate radicals in the sense that a radical class has a hereditary semisimple class if and only if it is an A-radical, i.e. the radical only depends on the structure of the underlying abelian groups (cf. [4]a and [5]). In the class of all near-rings (or abelian near-rings) the situation is not as bad, here we have both hereditary and non-hereditary semisimple classes, see, for example [3]. Dropping the associativity in this case, the result is even worse than in that of non-associative rings. We show that in the class of all abelian, not-necessatily associative zero-symmetric near-rings the only radicals with hereditary semisimple classes are the two trivial radical classes.

Original languageEnglish
Pages (from-to)273-275
Number of pages3
JournalNorth-Holland Mathematics Studies
Volume137
Issue numberC
DOIs
Publication statusPublished - 1987

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Near-ring
Semisimple
Associativity
Ring
Class
Abelian group
Trivial
If and only if
Alternatives

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Some Pathology for Radicals in Non-Associative Near-Rings. / Veldsman, Stefan.

In: North-Holland Mathematics Studies, Vol. 137, No. C, 1987, p. 273-275.

Research output: Contribution to journalArticle

Veldsman, Stefan. / Some Pathology for Radicals in Non-Associative Near-Rings. In: North-Holland Mathematics Studies. 1987 ; Vol. 137, No. C. pp. 273-275.
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