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Transactions of the American Mathematical Society


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 5^2, if A is an ideal in N, then A* is also an ideal. (N is called an snaring.) 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 (a), 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 (a) if and only if for each finitely generated subring A and each positive integer m, there is an integer f(m) such that Af(m)^Am. (Here A! = AS and Am + 1 = Asm.) Furthermore, conditions are given on the identities an s-naring N satisfies which will insure that N satisfies (a). It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.

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