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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Prime ideals in a large class of nonassociative rings

Author: Paul J. Zwier
Journal: Trans. Amer. Math. Soc. 158 (1971), 257-271
MSC: Primary 17.10
MathSciNet review: 0281763
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Abstract: In this paper a definition is given for a prime ideal in an arbitrary nonassociative ring $ N$ under the single restriction that for a given positive integer $ s \geqq 2$, if $ A$ is an ideal in $ N$, then $ {A^s}$ is also an ideal. ($ N$ is called an $ s$-naring.) This definition is used in two ways. First it is used to define the prime radical of $ N$ and the usual theorems ensue. Second, under the assumption that the $ s$-naring $ N$ has a certain property $ (\alpha )$, the Levitzki radical $ L(N)$ of $ N$ is defined and it is proved that $ L(N)$ is the intersection of those prime ideals $ P$ in $ N$ whose quotient rings are Levitzki semisimple. $ N$ has property $ (\alpha )$ if and only if for each finitely generated subring $ A$ and each positive integer $ m$, there is an integer $ f(m)$ such that $ {A^{f(m)}} \subseteq {A_m}$. (Here $ {A_1} = {A^s}$ and $ {A_{ m + 1}} = A_m^s$.)

Furthermore, conditions are given on the identities an $ s$-naring $ N$ satisfies which will insure that $ N$ satisfies $ (\alpha )$. It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.

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Keywords: $ s$-naring, prime ideal, semiprime ideal, prime radical, Levitzki radical, $ G$-system, $ G'$-system, $ G$-radical, $ G'$-radical, alternative ring, standard ring, generalized standard ring
Article copyright: © Copyright 1971 American Mathematical Society