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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: <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img32.gif" ALT="$s$">-naring, prime ideal, semiprime ideal, prime radical, Levitzki radical, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$G$">-system, <IMG WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0" SRC="images/img31.gif" ALT="$G’$">-system, <IMG WIDTH="22" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img30.gif" ALT="$G$">-radical, <IMG WIDTH="27" HEIGHT="22" ALIGN="BOTTOM" BORDER="0" SRC="images/img33.gif" ALT="$G’$">-radical, alternative ring, standard ring, generalized standard ring
Article copyright: © Copyright 1971 American Mathematical Society