Vol. 56, No. 1, 1975

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More on a generalization of commutative and alternative rings

Margaret Humm Kleinfeld

Vol. 56 (1975), No. 1, 159–170
Abstract

Let R be a ring such that in every subring generated by two elements the following identities are satisfied:

(x,y2,x) = y (x,y,x) (1)
(x,y,z) + (y,z,x) + (z,x,y) = 0 (2)
((x,y),x,x) = 0, (3)
where (a,b,c) = (ab)c a(bc),(a,b) = ab ba, and a b = ab + ba. This condition is satisfied by any alternative ring and also by any commutative ring. Assume further that R is a simple ring of characteristic not 2 or 3 and that R has an idempotent e such that (e,e,R) = 0 = (R,e,e) while (e,R)0. It is proved in this paper that under these conditions R must be alternative.

Mathematical Subject Classification 2000
Primary: 17A30
Milestones
Received: 7 August 1973
Published: 1 January 1975
Authors
Margaret Humm Kleinfeld