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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Prime ideals in a large class of nonassociative rings
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by Paul J. Zwier PDF
Trans. Amer. Math. Soc. 158 (1971), 257-271 Request permission

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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 257-271
  • MSC: Primary 17.10
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0281763-1
  • MathSciNet review: 0281763