The Radical of an Alternative Ring
1948; Princeton University; Volume: 49; Issue: 3 Linguagem: Inglês
10.2307/1969053
ISSN1939-8980
Autores Tópico(s)graph theory and CDMA systems
ResumoIn this paper we shall show that N. Jacobson's definition of the radical of an (associative) ring [J1, B] applies to alternative rings [Z1, M], and we shall develop some of the elementary properties of this radical. The radical of an alternative ring was first discussed by M. Zorn [Z2] under certain finiteness assumptions. Dubisch and Perlis [D P] have more recently studied the radical of an alternative algebra (of finite order). We make no finiteness assumptions. We do, however, prove that the chain conditions employed by Zorn [Z2, (4.2.1)(4.2.3)] ensure that the radical defined by him coincides with that defined in this paper. Our discussion applies equally well to algebras of possibly infinite order [JI, ?6] so that our results essentially contain some of the results of Dubisch and Perlis. We have been unable to discover the relation between the radical and maximal right ideals, nor have we developed any parallel for Jacobson's structure theory of associative rings. The enlarged radical of Brown and McCoy [B-McC] will yield a type of structure theory for arbitrary non-associative rings, but because of the generality involved we prefer to leave a discussion of this interesting fact to a subsequent publication.
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